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The combination formula is a powerful tool in probability and combinatorics used to find out how many ways we can select items from a larger set. It answers questions like "How many groups of a certain size can be chosen from a larger group?" The formula itself is given by:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Here, \(n\) represents the total number of items, and \(k\) is the number of items we want to choose. The exclamation mark, \(!\), denotes a factorial, which is the product of all positive integers up to that number.
For instance, to calculate the number of ways we can choose 3 members from a group of 12, we set \(n = 12\) and \(k = 3\). Plugging these into the formula, we get:\[\binom{12}{3} = \frac{12!}{3!(12-3)!} = 220\]This tells us there are 220 different ways to choose a group of 3 from 12. It’s important because we use this value as the denominator in probability calculations, representing all possible outcomes.

Probability is a way of quantifying the chance of an event occurring. It is a ratio of favorable outcomes to all possible outcomes, and is always a number between 0 and 1. In the context of our exercise, to find the probability that all members of the committee are men, we first need the number of ways to choose such a group.
We begin by identifying the number of favorable outcomes. Since there are 9 men on the board, we calculate how many ways we can select 3 men from these 9 using the combination formula:\[\binom{9}{3} = \frac{9!}{3!(9-3)!} = 84\]The probability that all members of the committee are men (our favorable event) is thus the number of favorable outcomes (84) divided by the total number of possible outcomes (220):\[P(\text{all men}) = \frac{84}{220} = \frac{21}{55} \approx 0.3818\]This gives us a probability of approximately 0.3818, indicating the likelihood that every committee member is male.

The complement rule is a useful probability tool that helps us find the probability of the opposite event occurring. If you know the probability of an event \(A\) happening, the probability of \(A\) not happening (complement of \(A\)) is 1 minus the probability of \(A\). This is written as:\[P(\text{not } A) = 1 - P(A)\]In our exercise, the complement rule is essential for finding the probability that at least one member of the committee is a woman. Instead of directly calculating this probability, which could be complex, we use the complement of this event, which is all members being men.
Having calculated that the probability all members are men is approximately 0.3818, we can now find:\[P(\text{at least 1 woman}) = 1 - P(\text{all men}) = 1 - 0.3818 \approx 0.6182\]This gives us a probability of about 0.6182 that there is at least one woman on the committee, demonstrating how effectively the complement rule simplifies certain probability calculations.