← Probability is favorable over total outcomes · Probability as Outcome Fraction

Probability is favorable over total outcomes · 12 practice problems

7.SP.C.77.SP.C.5

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling an even number
$㉡$ The probability of rolling a prime number
$㉢$ The probability of rolling a number that is $2$ or less

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving an even number are 2, 4, 6, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a prime number are 2, 3, 5, so that is 3 out of 6.

P(㉡) = \frac{1}{2}

3 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

2

or less are 1, 2, so that is 2 out of 6.

P(㉢) = \frac{1}{3}

2 of the six faces work.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

\frac{1}{3} < \frac{1}{2} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ covers 2 faces while the others cover more, so it is the smallest.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 2 hard answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling an odd number
$㉡$ The probability of rolling a number that is $2$ or less
$㉢$ The probability of rolling a number that is $6$ or greater

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving an odd number are 1, 3, 5, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

2

or less are 1, 2, so that is 2 out of 6.

P(㉡) = \frac{1}{3}

2 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

6

or greater are 6, so that is 1 out of 6.

P(㉢) = \frac{1}{6}

1 of the six faces work.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

\frac{1}{6} < \frac{1}{3} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ covers 1 face while the others cover more, so it is the smallest.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 3 medium answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling an odd number
$㉡$ The probability of rolling a number that is $5$ or greater
$㉢$ The probability of rolling a number greater than $6$

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving an odd number are 1, 3, 5, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

5

or greater are 5, 6, so that is 2 out of 6.

P(㉡) = \frac{1}{3}

2 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number greater than

6

are none, so that is 0 out of 6.

P(㉢) = 0

No face works, so this cannot happen.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

0 < \frac{1}{3} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ has no face at all, so its probability is 0 -- nothing can be lower.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 4 easy answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling an even number
$㉡$ The probability of rolling a multiple of $3$
$㉢$ The probability of rolling a number that is $6$ or greater

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving an even number are 2, 4, 6, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a multiple of

3

are 3, 6, so that is 2 out of 6.

P(㉡) = \frac{1}{3}

2 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

6

or greater are 6, so that is 1 out of 6.

P(㉢) = \frac{1}{6}

1 of the six faces work.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

\frac{1}{6} < \frac{1}{3} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ covers 1 face while the others cover more, so it is the smallest.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 5 medium answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling a prime number
$㉡$ The probability of rolling a number that is $2$ or less
$㉢$ The probability of rolling a number greater than $6$

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving a prime number are 2, 3, 5, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

2

or less are 1, 2, so that is 2 out of 6.

P(㉡) = \frac{1}{3}

2 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number greater than

6

are none, so that is 0 out of 6.

P(㉢) = 0

No face works, so this cannot happen.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

0 < \frac{1}{3} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ has no face at all, so its probability is 0 -- nothing can be lower.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 6 easy answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling an even number
$㉡$ The probability of rolling a number that is $2$ or less
$㉢$ The probability of rolling a number less than $1$

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving an even number are 2, 4, 6, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

2

or less are 1, 2, so that is 2 out of 6.

P(㉡) = \frac{1}{3}

2 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number less than

1

are none, so that is 0 out of 6.

P(㉢) = 0

No face works, so this cannot happen.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

0 < \frac{1}{3} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ has no face at all, so its probability is 0 -- nothing can be lower.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 7 hard answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling an odd number
$㉡$ The probability of rolling a number that is $6$ or greater
$㉢$ The probability of rolling a number less than $1$

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving an odd number are 1, 3, 5, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

6

or greater are 6, so that is 1 out of 6.

P(㉡) = \frac{1}{6}

1 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number less than

1

are none, so that is 0 out of 6.

P(㉢) = 0

No face works, so this cannot happen.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

0 < \frac{1}{6} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ has no face at all, so its probability is 0 -- nothing can be lower.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 8 easy answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling an odd number
$㉡$ The probability of rolling a multiple of $3$
$㉢$ The probability of rolling a number greater than $6$

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving an odd number are 1, 3, 5, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a multiple of

3

are 3, 6, so that is 2 out of 6.

P(㉡) = \frac{1}{3}

2 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number greater than

6

are none, so that is 0 out of 6.

P(㉢) = 0

No face works, so this cannot happen.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

0 < \frac{1}{3} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ has no face at all, so its probability is 0 -- nothing can be lower.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 9 medium answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling a multiple of $3$
$㉡$ The probability of rolling a prime number
$㉢$ The probability of rolling a number that is $6$ or greater

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving a multiple of

3

are 3, 6, so that is 2 out of 6.

P(㉠) = \frac{1}{3}

2 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a prime number are 2, 3, 5, so that is 3 out of 6.

P(㉡) = \frac{1}{2}

3 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

6

or greater are 6, so that is 1 out of 6.

P(㉢) = \frac{1}{6}

1 of the six faces work.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

\frac{1}{6} < \frac{1}{3} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ covers 1 face while the others cover more, so it is the smallest.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 10 medium answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling an even number
$㉡$ The probability of rolling a multiple of $3$
$㉢$ The probability of rolling a number less than $1$

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving an even number are 2, 4, 6, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a multiple of

3

are 3, 6, so that is 2 out of 6.

P(㉡) = \frac{1}{3}

2 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number less than

1

are none, so that is 0 out of 6.

P(㉢) = 0

No face works, so this cannot happen.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

0 < \frac{1}{3} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ has no face at all, so its probability is 0 -- nothing can be lower.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 11 hard answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling a multiple of $2$
$㉡$ The probability of rolling a number that is $5$ or greater
$㉢$ The probability of rolling a number greater than $6$

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving a multiple of

2

are 2, 4, 6, so that is 3 out of 6.

P(㉠) = \frac{1}{2}

3 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

5

or greater are 5, 6, so that is 2 out of 6.

P(㉡) = \frac{1}{3}

2 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number greater than

6

are none, so that is 0 out of 6.

P(㉢) = 0

No face works, so this cannot happen.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

0 < \frac{1}{3} < \frac{1}{2}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ has no face at all, so its probability is 0 -- nothing can be lower.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.

Variant 12 hard answer: ㉢

You roll a standard die with faces showing $1$ through $6$ . Among the following, find the one with the lowest probability.

$㉠$ The probability of rolling a multiple of $3$
$㉡$ The probability of rolling a number that is $5$ or greater
$㉢$ The probability of rolling a number less than $1$

Show solution

1 · Understandwhat's really being asked

Three events are described for a single roll of a fair six-sided die. We must say which is least likely.

Givens

The die is fair, with faces 1 through 6.
Three events are offered, one per marker.

Unknowns

Which of the three has the lowest probability.

Constraints

Every face is equally likely, so probability is just a count out of six.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Write out the six faces once, then for each event mark which faces belong to it. With the same denominator throughout, comparing is just comparing counts.

3 · Execute5 carry out the plan

1List the whole sample space

#2 Make a Systematic List 7.SP.C.7

A die has six faces, all equally likely, and every probability here is a count of those six.

\{1, 2, 3, 4, 5, 6\}

Six outcomes, each as likely as any other.

2Count and write ㉠

#2 Make a Systematic List 7.SP.C.7

The faces giving a multiple of

3

are 3, 6, so that is 2 out of 6.

P(㉠) = \frac{1}{3}

2 of the six faces work.

3Count and write ㉡

#2 Make a Systematic List 7.SP.C.7

The faces giving a number that is

5

or greater are 5, 6, so that is 2 out of 6.

P(㉡) = \frac{1}{3}

2 of the six faces work.

4Count and write ㉢

#2 Make a Systematic List 7.SP.C.7

The faces giving a number less than

1

are none, so that is 0 out of 6.

P(㉢) = 0

No face works, so this cannot happen.

5Compare the three fractions

#6 Guess and Check 7.SP.C.5

All three are sixths already, so the smallest numerator wins.

0 < \frac{1}{3} < \frac{1}{3}

㉢ is the least likely.

Answer: ㉢

4 · Reviewdoes it hold up?

㉢ has no face at all, so its probability is 0 -- nothing can be lower.

Another way: Shading the winning faces on a drawn die for each event shows the same ordering without writing any fractions.

Standardsmin grade 7

7.SP.C.7 Develop probability models and use them to find probabilities of events — Building each event's probability as favourable faces over six.
7.SP.C.5 Understand that the probability of a chance event is between 0 and 1 — Comparing the three probabilities on the 0-to-1 scale.

💡Takeaway. Probability on a fair die is just counting faces. An event no face satisfies has probability zero, not a small one.