i have no clue how to do this problem. any help would be appreciated!! :
Three hundred people apply for three jobs. 90 of the applicants are women.
(a) If three persons are selected at random, what is the probability that all are women? (Round the answer to six decimal places.)
(b) If three persons are selected at random, what is the probability that two are women? (Round the answer to six decimal places.)
(c) If three persons are selected at random, what is the probability that one is a woman? (Round the answer to six decimal places.)
(d) If three persons are selected at random, what is the probability that none is a woman? (Round the answer to six decimal places.)
(e) If you were an applicant, and the three selected people were not of your gender, should the above probabilities have an impact on your situation? Why?
Yes, the probabilities indicate the presence of gender discrimination.
No, because in the hiring process all outcomes are not equally likely.
so we have 210 men and 90 women,
Must choose 3
a) (90/300)(89/299)(88/298) = .026370
or
C(90,3)/C(300,3) = .026370
b) so you want permutations of WWM
= 3(90/300)(89/299)(210/298) = .188784
or
C(90,2)C(210,1)/C(300,3) = .188784
c) short way:
C(90,1)C(210,2)/C(300,3) = .....
d) none is woman ---> all 3 men
C(210,3)/C(300,3) = ...
e) your call.
Sure, I can help you with this problem! To solve it, we need to use the concept of probability.
(a) The probability of all three selected persons being women can be calculated by dividing the number of favorable outcomes (three women) by the total number of possible outcomes (any three people selected).
First, let's find the total number of possible outcomes. Since three persons are selected out of 300 applicants, we can calculate this using combinations. The number of combinations of 300 taken 3 at a time is given by the formula C(300, 3) = 300! / (3! * (300-3)!), where C(n, r) represents the number of combinations of n items taken r at a time.
So, the total number of possible outcomes is C(300, 3).
Next, let's find the number of favorable outcomes. Since there are 90 women out of 300 applicants, we want to find the number of combinations of 90 women taken 3 at a time, which is C(90, 3).
The probability of all three selected persons being women is then C(90, 3) / C(300, 3). Calculate this ratio to get the answer.
(b) The probability of two out of three selected persons being women can be calculated using the same method as in part (a). Now, we want to find the number of combinations of 90 women taken 2 at a time and multiply it by the number of combinations of 210 men taken 1 at a time. Then, divide this by the total number of possible outcomes C(300, 3).
(c) The probability of one out of three selected persons being a woman can be calculated in a similar manner. Now, we want to find the number of combinations of 90 women taken 1 at a time and multiply it by the number of combinations of 210 men taken 2 at a time. Then, divide this by the total number of possible outcomes C(300, 3).
(d) The probability of none of the three selected persons being women can be calculated by finding the number of combinations of 210 men taken 3 at a time divided by the total number of possible outcomes C(300, 3).
(e) Whether the above probabilities have an impact on your situation depends on the hiring process. If the hiring process is unbiased and all outcomes are equally likely, then the probabilities can provide an indication of the presence of gender discrimination. However, if the hiring process is not random and does not follow simple probability calculations, then the probabilities may not accurately reflect the situation. It's important to consider the specific circumstances of the hiring process to determine if gender discrimination is present.
To solve these problems, we will use probability formulas. Let's go step by step.
(a) To find the probability that all three selected people are women, we need to calculate the probability of selecting a woman on each of the three selections and multiply them together.
The probability of selecting a woman on the first choice is 90/300 = 3/10.
For the second choice, after one woman has already been selected, there are now 89 women left out of 299 applicants. So the probability of selecting a woman on the second choice is 89/299.
For the third choice, there are 88 women left out of 298 applicants. So the probability of selecting a woman on the third choice is 88/298.
To find the probability of all three events happening, we multiply the probabilities together:
P(all women) = (3/10) * (89/299) * (88/298)
You can compute this expression to find the answer.
(b) To find the probability that two out of the three selected people are women, we need to consider the different combinations of selecting two women and one man.
There are three possible scenarios:
1. Woman, Woman, Man,
2. Woman, Man, Woman,
3. Man, Woman, Woman.
We need to find the probability of each scenario happening and sum them up.
The probability of selecting a woman on the first choice is the same as in part (a): 3/10.
For the second choice, after one woman and one man have been selected, there are 88 women left out of 298 applicants. So the probability of selecting a woman on the second choice is 88/298.
And again, for the third choice, there are 89 women left out of 297 applicants. So the probability of selecting a woman on the third choice is 89/297.
Now, we multiply these probabilities for each scenario, sum them up, and round the answer to six decimal places.
(c) To find the probability that one out of the three selected people is a woman, we need to consider the different combinations of selecting one woman and two men.
There are three possible scenarios:
1. Woman, Man, Man,
2. Man, Woman, Man,
3. Man, Man, Woman.
The probability of selecting a woman on the first choice is the same as in part (a): 3/10.
For the second choice, after one woman has been selected, there are 210 men left out of 299 applicants. So the probability of selecting a man on the second choice is 210/299.
And again, for the third choice, there are 209 men left out of 298 applicants. So the probability of selecting a man on the third choice is 209/298.
Now, we multiply these probabilities for each scenario, sum them up, and round the answer to six decimal places.
(d) To find the probability that none of the three selected people are women, we need to calculate the probability of selecting a man on each of the three choices.
The probability of selecting a man on the first choice is 210/300 = 7/10.
For the second choice, after one man has already been selected, there are now 209 men left out of 299 applicants. So the probability of selecting a man on the second choice is 209/299.
And again, for the third choice, there are 208 men left out of 298 applicants. So the probability of selecting a man on the third choice is 208/298.
Now, we multiply these probabilities together and round the answer to six decimal places.
(e) The above probabilities should have an impact on your situation because they indicate the presence of gender discrimination. If the probabilities for certain genders are significantly different from each other, it suggests that the hiring process might not be fair and equal. This information can be taken into consideration when assessing the fairness and inclusivity of the selection process.