In this video, we'll continue our introduction to probability. Our learning objectives are to discriminate between marginal, joint, and conditional probabilities, to discriminate between independent and dependent events, and to calculate marginal, joint, and conditional probability under independent and dependent conditions. There's a difference between statistical independence and dependence. Events which are statistically independent are those where the outcome of one event has no effect on the outcome of the second event. Events which affect subsequent events are termed dependent. Next, we'll look at probabilities under independent conditions. First, we'll start with marginal probability. This is simply the single probability of an independent event, for example, a coin toss. With the coin toss, I'm going to get either heads or tails, which is one side or the other, and the probability that I get heads or the probability I get tails is going to be 50 percent. Under independent conditions I can also have joint probability, which is the probability of two or more events occurring together or in succession and that is the product of their marginal probabilities. The probability of A and B is the probability of A times B, where the probability of AB is the probability of events A and B occurring together or in succession, in other words, the joint probability. Probability of A is the marginal probability of A and the probability of B is the marginal probability of B. Let's take a look at an example. The probability of machine operator producing a defective part at any point in time is 0.05. What is the probability that three bad parts will be produced in succession or three defective parts? The probability that I get a defective and then another defective and then another defective or the joint probability of those three events, the probability of A, B, and C is the probability of A times the probability of B times the probability of C. In this case, the probability is consistent for events, A, B and C. The probability of getting three defectives in a row would be the probability of getting a defective times the probability of getting a defective or 0.5 times 0.5 times 0.5, which would yield a result of 0.000125. We can also have conditional probability under independent conditions. Now, there's an interesting trick here. The probability of B given that A has occurred is written as such, probability of B and this upward slash and then A after it. The probability of B given A has occurred is simply going to be the probability of B. The reason why is because A and B are independent and B doesn't influence A or vice versa. When we have dependent conditions, the conditional probability is different. The probability of B given that A has occurred is the probability or the joint probability of A and B divided by the probability of A. Now, we'll look at an example here, but you'll note that this is equivalent to calculating a probability of the part being defective given a sample space B after A has been drawn. Let's take a look of an example here. Assume a randomly selected part is from vendor A, what is the probability that it is also defective? Now, we're looking at the probability defective given that A has already occurred. You can see I've reduced my sample space down to vendor A. I want to see what the probability of this also defective. The probability that it's defective part given that it's from vendor A is the same as the joint probability of defective and from vendor A divided by the probability that is from vendor A. We'll see that the joint probability of defective and from vendor A if we go back to our table, we'd look at vendor A and defective and that's 15 out of a total of 165. Then I'll divide that by the probability of A or probability of vendor A, which the probability of it being from vendor A is the marginal probability. You'll notice we call it marginal because it's over here in the margin. That is 100/165 total and that's equal to 0.15. Now, you'll note this is the same as observing that given the 15 defectives out of 100 vendor A parts, 15 out of 100 is 0.15. Note also that the probability of a part being defective and from vendor A constitutes a joint probability under statistical dependence. We can create a table of joint probability values for the sample space, where we have the event that the parts are from vendor A and defective which is 15 out of 165. We have the event that it's from vendor A and not defective and probability of 85 out of 165. We have the event for vendor B and defective and that's a total of 165 and vendor B not defective at 55 out of 165. Now, as a second example, assume that we had a non defective part that had been drawn. What's the probability that it's from vendor B? The probability of vendor B given that we already reduced this sample space to the non defectives is the joint probability of vendor B and not defective, which, as we can see here, is 0.333 divided by the probability that it's from the non defective group, and of course, we go back here and that'd be140/165. We get a probability of 0.3929. Now, note that should a non defective part have been selected, the probability of it being a part from vendor B is 55 out of 140. Again, if we go back and we reduce it to this sample space right here being the not defective group, the probability of this vendor B would be 55/140. Now, for joint probabilities under statistical dependence there's a formula and it's a variation of the conditional probability formula. You can see the formulas is here. The probability of B given that event A has occurred is the joint probability of B and A divided by the probability of A. If we want to get the joint probability of A and B or B and A, it's the probability of B given A has occurred times the probability of A. Now, note that the joint probability of B and A is the same as the joint probability of A and B. This probability is either happening together or in succession. As an example, we can check any of the joint probability calculations from the joint probability table. For example, the probability of A and defective is 0.0909 or 15 out of 165. Now, if we go back to the formula here, we can look at it and using our formula that we just talked about, the probability of A and defective divided by the probability of defective times the probability defective, of course these cross out, but we'll take that and you'll see that these in the denominator in the numerator, these cross out, and we're left with the joint probability.
