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In a company there are 99 executives: 33 women and 66 men. Three are selected at random to attend a management seminar. Find these probabilities. All three selected will be women. Round your answer to five decimal places. The probability that all three people selected will be women is

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Q. In a company there are 99 executives: 33 women and 66 men. Three are selected at random to attend a management seminar. Find these probabilities. All three selected will be women. Round your answer to five decimal places. The probability that all three people selected will be women is
  1. Calculate Total Combinations: First, calculate the total number of ways to choose 33 executives out of 99, regardless of gender. This is a combination problem, so use the formula for combinations: nCr=n!r!(n−r)!nCr = \frac{n!}{r!(n-r)!}, where nn is the total number of items, rr is the number of items to choose, and ＂!!＂ denotes factorial.
  2. Calculate Women Combinations: Calculate the total combinations: 9C3=9!(3!(9−3)!)=9!(3!6!)=(9×8×7)(3×2×1)=849C3 = \frac{9!}{(3!(9-3)!)} = \frac{9!}{(3!6!)} = \frac{(9\times8\times7)}{(3\times2\times1)} = 84.
  3. Calculate Probability: Now, calculate the number of ways to choose 33 women out of the 33 available. This is also a combination problem: 3C3=3!(3!(3−3)!)=3!(3!0!)=13C3 = \frac{3!}{(3!(3-3)!)} = \frac{3!}{(3!0!)} = 1.
  4. Calculate Probability: To find the probability that all three selected will be women, divide the number of ways to choose 33 women by the total number of ways to choose 33 executives: Probability = Number of ways to choose 3 womenTotal number of ways to choose 3 executives\frac{\text{Number of ways to choose } 3 \text{ women}}{\text{Total number of ways to choose } 3 \text{ executives}}.
  5. Round Probability: Calculate the probability: Probability=184\text{Probability} = \frac{1}{84}.
  6. Round Probability: Calculate the probability: Probability=184\text{Probability} = \frac{1}{84}. Round the answer to five decimal places: Probability≈0.01190\text{Probability} \approx 0.01190.

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