A committee of three is to be chosen from a group of four women and three men. What is the probability that the committee will consist of one man and two women?

Question

BlackTom AI · Accepted Answer

First, restate the problem in our own words: We need the probability that a 3-person committee drawn from 4 women and 3 men consists of exactly 1 man and 2 women.

Option a describes a value that clearly does not match the combinatorial calculation, since it suggests a ratio that would be far from the true count based on combinations; without computing, you can sense it’s unlikely to be a clean 18 out of 35 proportion.

To compute the correct probability, count the favorable committees: choose 1 man from 3 and 2 women from 4. This gives C(3,1) ways for the men and C(4,2) ways for the women. Multiply: C(3,1) × C(4,2) = 3 × 6 = 18 favorable committees.

Next, count the total number of ways to form any 3-person committee from the 7 people (4 women + 3 men). That is C(7,3) = 35 total committees.

Therefore, the probability is (favorable)/(total) = 18/35.

Now evaluate each option:

- Option a: If it represents 135 over 35 or any equivalent large ratio, it does not reduce to the correct 18/35. The structure 135/35 would simplify to 27/7, which is not in the [0,1] probability range that 18/35 resides in, so this cannot be correct.
- Option b: A mixed number like 16 1/6 is 97/6, which is far greater than 1 and thus impossible as a probability; probabilities must be between 0 and 1, so this cannot be correct.
- Option c: A mixed number such as 37 3/7 similarly yields a value well above 1; it cannot be a probability for a 3-member selection.
- Option d: The form 18/35 matches the computed probability exactly; the fraction reduces to a legitimate probability between 0 and 1, and 18 and 35 come directly from the combinatorial counts above.
- Option e: The notation 12 1/2 (or any form resembling 12.5) is far outside the probability range [0,1], so this is not correct.

In summary, the valid probabilistic value for this scenario is the fraction 18/35, which corresponds to the count of favorable committees over the total number of committees.

A committee of three is to be chosen from a group of four women and three men. What is the probability that the committee will consist of one man and two women?单项选择题

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