Final answer:
The probability of randomly selecting all 3 women from a group of 9 sales representatives (6 men and 3 women) is approximately 1.19%. This is based on a hypergeometric distribution. While low probability does not prove non-randomness, it raises questions about the selection process.
Step-by-step explanation:
When evaluating whether there is reason to doubt the manager's claim that the 3 people (all women) were selected at random from a group of 9 sales representatives (6 men and 3 women) to attend a national convention, we can use probability to assess the likelihood of this occurring.
Since there are only 3 women, and 3 people needed to be selected, the question essentially asks what the probability is of selecting all 3 women from the overall group of 9 people.
This is an example of a hypergeometric distribution, where we are selecting without replacement from two distinct groups (in this case, men and women).
To calculate the probability of selecting all 3 women, we could use the hypergeometric probability formula:
P(X = 3) = [(C(3,3) * C(6,0)) / C(9,3)]
Where C(n, k) represents the combinations of k items from n items. However, since C(3,3) is 1 and C(6,0) is also 1, the calculation simplifies to:
P(X = 3) = 1 / C(9,3) = 1 / 84, which is approximately 0.0119 or 1.19%.
The calculation indicates that the probability of this event occurring by chance is very low.
While a low probability does not prove that the process was not random, it does suggest that the outcome is quite rare and might raise some questions regarding the selection process.