SAT Probability Question 29 | Free SAT Prep | Aniko

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Question 29·Medium·Probability and Conditional Probability

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Two fair six-sided dice are rolled. What is the probability that the sum of the dice is at least 9 given that at least one of the dice shows a 5?

For conditional probability questions with dice or cards, always shrink the sample space first using the condition (here, "at least one die shows a 5"). List or systematically count only those outcomes that satisfy the condition, then within that smaller set count the ones that meet the target event (such as a minimum sum). Finally, form the probability as favorable over total in the reduced sample space and simplify; this structured counting avoids guesswork and common double-counting mistakes.

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Use ordered pairs and the condition

Represent each roll as an ordered pair like $(3,5)$ , where the first number is die 1 and the second is die 2. Because of the phrase "given that at least one of the dice shows a 5," you should ignore any outcome that does not include a 5.

Count the reduced sample space

How many ordered pairs have at least one 5? Try counting outcomes where the first die is 5, outcomes where the second die is 5, and then be careful about any overlap you might have counted twice.

Identify favorable outcomes within that space

Once you know all the outcomes that include a 5, go through that list and mark which of them have sums of 9 or more. Then the probability is the number of marked outcomes divided by the total number of outcomes that include a 5.

Form the final fraction

Your final step is to take (number of outcomes with at least one 5 and sum at least 9) over (number of outcomes with at least one 5). Simplify that fraction if possible.

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Use Desmos to check your fraction

After you work out, on paper, how many outcomes satisfy the condition and how many of those have sum at least 9, type your fraction (favorable ÷ total under the condition) into Desmos. Compare the decimal that Desmos shows to the decimal forms of the answer choices to confirm which one matches your result.

Step-by-step Explanation

Focus on the conditional event

We are asked for a conditional probability: the probability that the sum is at least 9 given that at least one die shows a 5.

This means we only consider outcomes where at least one die is 5, and then find what fraction of those have a sum of at least 9.

Think of each roll as an ordered pair $(\text{die}_1,\text{die}_2)$ , like $(2,5)$ or $(5,6)$ .

List or count all outcomes with at least one 5

Total outcomes when rolling two dice: $6\times 6 = 36$ .

Now count the outcomes where at least one die is 5.

First die is 5: $(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)$ → 6 outcomes.
Second die is 5: $(1,5),(2,5),(3,5),(4,5),(5,5),(6,5)$ → 6 outcomes.

We counted $(5,5)$ twice, so subtract 1:

Total with at least one 5: $6 + 6 - 1 = 11$ outcomes.

These 11 outcomes form our new sample space.

Find which of those outcomes have sum at least 9

From the 11 outcomes with at least one 5, list their sums:

$(5,1): 6$ , $(5,2): 7$ , $(5,3): 8$ , $(5,4): 9$ , $(5,5): 10$ , $(5,6): 11$
$(1,5): 6$ , $(2,5): 7$ , $(3,5): 8$ , $(4,5): 9$ , $(6,5): 11$

Now keep only those with sum at least 9:

$(5,4)$ → 9
$(5,5)$ → 10
$(5,6)$ → 11
$(4,5)$ → 9
$(6,5)$ → 11

So there are 5 favorable outcomes.

Compute the conditional probability

The conditional probability is

\text{Probability} = \frac{\text{favorable outcomes}}{\text{outcomes with at least one 5}} = \frac{5}{11}.

So the correct answer is $\boxed{\tfrac{5}{11}}$ .

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Step-by-step Explanation

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Step-by-step Explanation