An wrn is an opaque vessel whose contents cannot be seen....

An wrn is an opaque vessel whose contents cannot be seen. Three such ums $\left(A_1, A_2, A_3\right)$ each contain four balls. The balls are identical except for color: $A_1$ has three blue balls and one yellow, $A_2$ has two blue and two yellow, and $A_3$ has one blue and three yellow. The experiment is to randomly select one of the urns and then, without looking, select one ball from that um. Let $A_1, A_2, A_3$ represent the events of selecting a ball from the given um, $B$ represent the event of selecting a blue ball, and $B^{\prime}$ the event of selecting a yellow ball. From this information develop a joint probability table.

Key Concepts

Probability Tables

Probability tables are a tabular method used for organizing and displaying the probabilities of different outcomes for one or more random variables. They offer a clear visual layout that helps in calculating marginal, joint, and conditional probabilities efficiently.

Marginal Probability

Marginal probability refers to the probability of an event occurring irrespective of the outcomes of other variables. It is obtained by summing the joint probabilities over the other variable(s), and provides insights into the overall likelihood of a single event from a joint distribution.

Law of Total Probability

The law of total probability provides a way to calculate the probability of an event by considering all the different mutually exclusive scenarios that can produce the event. It breaks down the overall probability into a sum of probabilities across a partition of the sample space, each weighted by the probability of the condition.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. It is a fundamental concept used to update probabilities based on additional information, and is essential when outcomes depend on prior conditions.

Joint Probability Distribution

A joint probability distribution represents the probability that two or more events occur simultaneously. It organizes the probabilities of all combinations of outcomes from two or more random variables, allowing the assessment of how these variables interact with each other.

Transcript

00:02 Okay, for this problem they give you three balls that you're pulling out of an urn, b1, b2, and w.

00:12 And then they ask you to determine all the different possible ways that we can pull two of these out with replacement.

00:21 With replacement means the ball is put back in and can be pulled out again.

00:25 So if we list all the possibilities, what i'll do is i'll just do b1 in the beginning as my first ball and list all the combinations with it.

00:32 It could be b1 and b1 where that ball is pulled out twice.

00:35 It could be b1 with b2 or it could be b1 with w.

00:40 Those are my three possibilities with b1 being the first ball.

00:45 It could also be b2 being the first ball and we list all its different combinations.

00:49 B2 with b1, b2 with b2 pulling out the same ball twice, b2 with w.

00:57 And then we do the same thing with w.

00:59 We say w with b1, w with b2, and w with w.

01:05 Those are your nine possibilities.

01:07 That is the answer for part a.

01:10 And then for part b, they ask us to see what's the probability that two balls are drawn both blue.

01:19 And so for that situation, what we have to do is look at these different possibilities and say how many different ways can we get both blue.

01:25 Well, we have them up here.

01:27 Let's list them again...

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