Let X be the winnings of a gambler. Let p(i)= P(X=i) and suppose that p(0)=1 / 3 ; p(1)=p(-

Let X be the winnings of a gambler. Let p(i)= P(X=i) and...

Key Concepts

Normalization of Probabilities

Normalization in probability ensures that the total probability of all possible outcomes sums to 1. When computing conditional probabilities, the probabilities of the outcomes in the restricted sample space must sum to 1. In this example, after identifying the probabilities for winning positive amounts, each individual probability is normalized by dividing by the total probability of winning a positive amount.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. It is computed by taking the probability of the intersection of the events and dividing it by the probability of the conditioning event. In this problem, the probability that the gambler wins a specific positive amount is determined by dividing the probability of that specific winning by the total probability of winning any positive amount.

Discrete Probability Distribution

A discrete probability distribution specifies the probability for each value that a discrete random variable can take. In this context, the winnings of a gambler are modeled as a discrete random variable with specified probabilities for each outcome (e.g., winning -1, 0, 1, etc.). This forms the basis for calculating further probabilities and expectations about the outcomes.

Transcript

00:02 Okay, so for what i would do for this one is first let's just organize what they're giving us as a probability distribution.

00:08 We're going to create a little table organizing the information they're giving us.

00:11 They give us all these different i values that we're going to be finding the probability of and then we're going to go ahead and write down what the probability is for each situation.

00:21 So if we look at this carefully, they give us numbers anywhere from i equals negative three all the way up to i is equal to positive three.

00:32 And then we have to read the directions carefully.

00:34 They tell us the probability of zero is one third.

00:36 So i'm going to put that right there.

00:39 That's what this bottom row is for as the probabilities.

00:42 They tell us the probability of one and negative one are the same and they're both 13 over 55.

00:46 So i'm going to put a 13 over 55 here and another 13 over 55 here.

00:53 They tell us the probability of two and negative two are also the same and they tell us those values are one over 11.

00:58 So we put a one over 11 here and a one there and then the probability for three and negative three.

01:04 They're both one over 165.

01:14 Now to be a true probability distribution, all these probabilities we add them up equal to one and you could check that yourself.

01:21 I did plug them all in.

01:22 It does add up to one and that means this verifies this is a true probability distribution.

01:28 Now what they want us to do is do this conditional probability.

01:32 We have to read the directions carefully.

01:35 They say compute the conditional probability that the gambler wins.

01:38 That is i equals one two or three given that he wins a positive amount.

01:45 Okay, so we want to know the probability that he wins with a one given that we already know that he has won a positive amount.

02:01 And what we need to know is how do we do conditional probability? well, what we do here is the bottom is going to be the probability of the given which will be taking all of these probabilities right here.

02:20 We're disregarding anything that's not a positive amount and we're given that he's already won a positive amount.

02:26 And then what's the probability of winning with the one? well, we put the probability of the one at the top from this table right here.

02:34 So if we do that, we'll find the probability of him winning one dollar given that he's already won a positive amount.

02:41 That will be 13 over 55 over these three numbers added together.

02:54 Okay? so the way you do conditional probability is you put the probability of the given on the bottom of all the scenarios that we already know are true and then the probability of the one situation that we want to happen.

03:05 And we will figure that out and then we'll get an answer.

03:09 And when you add all these numbers up on the bottom, you get a value of one third there...

Please give Ace some feedback