A campus student club distributed material about membership...

A campus student club distributed material about membership to new students attending an orientation meeting. Of those receiving this material $40 \%$ were men and $60 \%$ were women. Subsequently, it was found that $7 \%$ of the men and $9 \%$ of the women who received this material joined the club. a. Find the probability that a randomly chosen new student who receives the membership material will join the club. b. Find the probability that a randomly chosen new student who joins the club after receiving the membership material is a woman.

A campus student club distributed material about membership to new students attending an orientation meeting. Of those receiving this material $40 \%$ were men and $60 \%$ were women. Subsequently, it was found that $7 \%$ of the men and $9 \%$ of the women who received this material joined the club.
a. Find the probability that a randomly chosen new student who receives the membership material will join the club.
b. Find the probability that a randomly chosen new student who joins the club after receiving the membership material is a woman.

Key Concepts

Total Probability Theorem

This theorem allows us to calculate the probability of an event by decomposing it into several disjoint cases whose probabilities are easier to work with. In situations where overall probabilities are determined by a mixture of subpopulations, the theorem provides a way to combine the conditional probabilities of each category with the probability of each category’s occurrence.

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It is essential for situations where the outcome or probability of one event depends on the occurrence of another, and is the foundation for updating probabilities based on new information.

Bayes' Theorem

Bayes' Theorem is a method for updating our probability estimates for events based on new or additional information. When data is partitioned across different groups, this theorem is employed to determine the probability that an occurrence belongs to a particular group given the overall observation.

Weighted Average Probability

Weighted average probability involves calculating an overall probability by accounting for different probabilities across distinct groups or scenarios, each multiplied by a factor representing the proportion of that group. This method is often used in conjunction with the total probability theorem in order to combine results from disparate subsets.

Transcript

00:01 Now, here in this problem, we are told that people receiving the material, 40 % were men.

00:04 So i'm just going to say probability of a man is 40%.

00:09 So that means the probability of a woman is 60%.

00:16 Now, we're also told that 7 % of the men who received the material joined the club.

00:24 That's what means the probability of joining, given that they were a man.

00:29 So we know that they were a man.

00:33 0 .07.

00:35 And then the probability of joining, given that they were a woman.

00:40 That was 9%.

00:41 So that would be 0 .09.

00:47 Part a, we want to find the probability that randomly chose a new student who receives the membership will join the club.

00:53 So here on part a, we just want to find the probability of just j.

01:01 Now let's use what we have above first.

01:05 Now this top line here tells us, using our rules of conditional probability, the probability of joint and m over the probability of a male is 0 .07.

01:17 Probability of male is 0 .40, so we can make that bottom 0 .40.

01:24 And then if we cross multiply here, this tells us the probability of joining and a man is equal to 0 .40 times 0 .07, which is 0 .028.

01:39 It's used a similar method here.

01:41 We need the probability of joining and a woman over the probability of a woman is 0 .09...

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