Video Transcript
In a given sample space 𝑆, the probabilities are shown for combinations of events 𝐴 and 𝐵 occurring. Are 𝐴 and 𝐵 independent events?
When two events are independent, the probability of one event occurring in no way affects the probability of the other event occurring. And we know something about the intersection of two independent events. On a Venn diagram, the probability of the intersection of 𝐴 and 𝐵 is the overlapping portion of 𝐴 and 𝐵. If events are independent, the probability of 𝐴 and 𝐵 both occurring will be equal to the probability of 𝐴 times the probability of 𝐵. For example, a die roll and a coin toss are independent events. The coin toss cannot affect the outcome of the die roll. And that means the probability of rolling a two on the die roll and flipping heads on the coin toss will be equal to one-sixth times one-half.
We don’t have a context or know what specific events 𝐴 and 𝐵 are, but we can use this principle to check and see if these events are independent. We know that the intersection of 𝐴 and 𝐵 is three twentieths. The probability of 𝐴 occurring is seven twentieths. It’s the probability that only 𝐴 occurs plus the probability that 𝐴 and 𝐵 occur. The probability of 𝐵 is then six twentieths. It’s the probability only 𝐵 occurs plus the probability that 𝐴 and 𝐵 occur.
Before we multiply these two probabilities together, we can simplify. Six twentieths reduces to three-tenths. Seven times three is 21. 20 times 10 is 200. And we recognize that three twentieths is not equal to 21 over 200. For this diagram in this sample space, the probability of the intersection of 𝐴 and 𝐵 is not equal to the probability of 𝐴 times the probability of 𝐵. And since that is not true, it must be the case that the probability of one event occurring influences the likelihood of the other event. Therefore, 𝐴 and 𝐵 are not independent events.
