A conference began at noon with two parallel sessions. The...

A conference began at noon with two parallel sessions. The session on portfolio management was attended by $40 \%$ of the delegates, while the session on chartism was attended by $50 \%$. The evening session consisted of a talk titled "Is the Random Walk Dead?" This was attended by $80 \%$ of all delegates. a. If attendance at the portfolio management session and attendance at the chartism session are mutually exclusive, what is the probability that a randomly chosen delegate attended at least one of these sessions? b. If attendance at the portfolio management session and attendance at the evening session are statistically independent, what is the probability that a randomly chosen delegate attended at least one of these sessions? c. Of those attending the chartism session, $75 \%$ also attended the evening session. What is the probability that a randomly chosen delegate attended at least one of these two sessions?

A conference began at noon with two parallel sessions. The session on portfolio management was attended by $40 \%$ of the delegates, while the session on chartism was attended by $50 \%$. The evening session consisted of a talk titled "Is the Random Walk Dead?" This was attended by $80 \%$ of all delegates.
a. If attendance at the portfolio management session and attendance at the chartism session are mutually exclusive, what is the probability that a randomly chosen delegate attended at least one of these sessions?
b. If attendance at the portfolio management session and attendance at the evening session are statistically independent, what is the probability that a randomly chosen delegate attended at least one of these sessions?
c. Of those attending the chartism session, $75 \%$ also attended the evening session. What is the probability that a randomly chosen delegate attended at least one of these two sessions?

Key Concepts

Mutually Exclusive Events

This concept refers to events that cannot occur simultaneously. In probability calculations, if two events are mutually exclusive, the probability of either event occurring is simply the sum of their individual probabilities. This principle is pivotal when there is no overlap between events, simplifying the computation of union probabilities.

Statistical Independence

Statistical independence between two events means that the occurrence of one event does not affect the occurrence of the other. When events are independent, the probability of both events occurring together is the product of their individual probabilities. This concept is essential for correctly calculating combined probabilities in cases where events occur without influencing each other.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is used to calculate the probability of the union of events by accounting for the overlap between them. In cases where events are not mutually exclusive, it prevents double-counting by subtracting the probability of their intersection after adding the individual probabilities. This principle is key to accurately determining the likelihood of at least one event occurring.

Conditional Probability

Conditional probability deals with the likelihood of an event occurring given that another event has already occurred. It provides a framework for updating probabilities based on new or shared information about events. This is particularly important when analyzing scenarios where the occurrence of one event impacts the probability of another.

Transcript

00:01 So we're given information about vegetarians or vegetarian meals.

00:06 That's 35%.

00:07 Probability that someone is a student is 0 .5.

00:12 And the probability that given that someone is a student that they order a vegetarian meal is 0 .25.

00:20 And we want to find in a, what is the probability of being a student and ordering vegetarian? and so we can either find the probability of a student and then followed by finding the probability of a vegetarian given a student or we find the probability of a vegetarian times the probability that someone is a student given that they're a vegetarian and the information we have is this conditional so we need to do it this way.

00:51 So we're going to take that point five times the point two five and that's going to give us a half of .25 is .125.

01:00 So there's our intersection.

01:03 Now part b asks us to find the probability of a student, given that someone orders a vegetarian meal, and that's going to be the probability of s and v, divided by the probability of being v.

01:19 So we have our intersection that we just found, 0 .125.

01:23 So if we messed up on that, this one's going to also be wrong, and then divided by .35.

01:30 And when we take .125 divided by .35, we get not a great answer, but .37, we will give it to four decimal places.

01:43 Now, we want to find part c, we want to find what's the probability that someone is not a student and they're not vegetarian.

01:53 And you know how i'm going to do this? i'm going to go through and draw a venn diagram because i want you to see how drawing venn diagrams is really, really helpful.

02:01 And here are the two circles.

02:03 Here's a vegetarian, here's student, and here's their intersection.

02:06 And we know this intersection is point.

02:10 I'll get that to click.

02:11 Point one, two, five...

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