Digital ACT Math Practice Question #1170
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This is a conditional probability problem. Start by identifying the condition that restricts your sample space. What are all the possible ways to select 3 books such that at least one non-fiction biography is included?
Once you've established your new, reduced sample space (the denominator for your probability), determine how many outcomes within that space satisfy the specific event you're interested in: selecting exactly 2 historical fiction novels (the numerator).
⚡ **Focus on the Reduced Sample Space:** For conditional probability problems, instead of using the formula P(B|A) = P(A and B) / P(A), which involves calculating probabilities relative to the total sample space, directly calculate the number of favorable outcomes within the *reduced* sample space defined by the condition. This often simplifies calculations by avoiding fractions until the very last step. 1. **Identify the condition (Event A):** 'at least one non-fiction biography'. 2. **Calculate the size of the reduced sample space:** * Total ways to pick 3 books from 10: C(10, 3) = 120. * Ways to pick 3 books with *no* non-fiction (i.e., all 3 historical fiction): C(6, 3) = 20. * Number of ways to pick 3 books with *at least one* non-fiction = Total ways - Ways with no non-fiction = 120 - 20 = 100. This is your new denominator. 3. **Identify the desired outcome (Event B) within this reduced sample space:** 'exactly 2 of the selected books were historical fiction novels'. * If 2 are historical fiction, and 3 books are selected in total, then 1 must be non-fiction. * Number of ways to pick 2 historical fiction from 6: C(6, 2) = 15. * Number of ways to pick 1 non-fiction from 4: C(4, 1) = 4. * Number of ways to pick exactly 2 HF and 1 NF = C(6, 2) * C(4, 1) = 15 * 4 = 60. This is your new numerator. 4. **Calculate the probability:** (Favorable outcomes) / (Reduced sample space) = 60 / 100 = 3/5. This method avoids fractions until the very end, reducing potential calculation errors.
This is a conditional probability problem. We need to find the probability of selecting exactly 2 historical fiction novels (Event B) GIVEN that at least one non-fiction biography was selected (Event A). We can solve this using the formula ...
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