The Probability of None Being Selected in a Job Interview Scenario

In a job interview scenario, two candidates (a man and a woman) apply for two positions. The probability of the man being selected is 1/4, while the probability of the woman being selected is 1/3. This article will explore the probability that neither of them will be selected.

Calculating the Probability

To find the probability that none of them will be selected, we first calculate the probabilities that each individual is not selected and then multiply these probabilities together. The events are independent, meaning the probability that neither is selected is the product of their individual probabilities of not being selected.

Probability of the Man Not Being Selected

Probability that the man is not selected:

P(man not selected)  1 - P(man selected)                     1 - frac{1}{4}                     frac{3}{4}

Probability of the Woman Not Being Selected

Probability that the woman is not selected:

P(woman not selected)  1 - P(woman selected)                       1 - frac{1}{3}                       frac{2}{3}

Probability that Neither of Them Will Be Selected

Since the events are independent, the probability that neither is selected is the product of their individual probabilities of not being selected:

P(none selected)  P(man not selected) * P(woman not selected)                  frac{3}{4} * frac{2}{3}                  frac{3 * 2}{4 * 3}                  frac{6}{12}                  frac{1}{2}

Therefore, the probability that none of them will be selected is (frac{1}{2}).

The Probability That Both Are Selected

The probability that both are selected is (frac{1}{7} times frac{1}{5} frac{1}{35}). The probability that neither is selected is (1 - frac{1}{7} - frac{1}{5} frac{6}{7} - frac{4}{35} frac{24}{35}).

The Probability That Exactly One Is Selected

The probability that exactly one is selected is (1 - frac{24}{35} - frac{1}{35} frac{35 - 25}{35} frac{10}{35} frac{2}{7}).

Conditional Probabilities and Independence

Let's denote the following probabilities:

PH  frac{4}{6}  PH'  frac{2}{6}PW  frac{3}{4}  PW'  frac{1}{4}

The probability that only one of them is selected (either the husband or the wife, but not both) is calculated as:

P(only 1 is selected)  PH * PW'   PH' * PW                       frac{4}{6} * frac{1}{4}   frac{2}{6} * frac{3}{4}                       frac{4}{24}   frac{6}{24}                       frac{10}{24}                       frac{5}{12}

Probability of Not Getting Selected

The probability of the wife not getting selected is (frac{4}{7}) and the husband not getting selected is (frac{1}{3}). The probability of neither of them not getting selected is (frac{1}{3} times frac{4}{7} frac{4}{21}) or approximately 19.05%.

When the events of getting selected or not getting selected are the only two options, their total makes 1. Therefore, the probability of not getting selected is 1 minus the probability of getting selected. For the wife, it is (1 - frac{3}{7} frac{4}{7}), and for the husband, it is (1 - frac{1}{3} frac{2}{3}).

The probability of both not being selected is (frac{1}{3} times frac{4}{7} frac{4}{21}), or around 19%.