True or False? In Exercises 3-6, determine whether the stateme...

Question

True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false,

explain why.

4. When two events are independent, they are also mutually exclusive.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says which of the following statements is correct regarding the relationship between independent and mutually exclusive events. We have 4 possible choices as our answers. For choice A, if two events are independent, they must be mutually exclusive. For choice B, if two events are mutually exclusive, they must be independent. For choice C. Two events can be both independent and mutually exclusive, if at least 1 has probability zero, and for choice D, 2 events are always independent if they are not mutually exclusive. So, in this problem, we're looking at independent and mutually exclusive events. So let's recall those definitions. So, for mutually exclusive events, we call that these are events that cannot happen at the same time. And the way we write that in terms of probabilities is that the probability, if we have two events, A and B, that the probability of a intersect B. is going to be equal to 0. So this is just saying that the probability of both event A and event B occurring at the same time as 0 if they are mutually exclusive events, since they both cannot occur at the same time. The next definition that we need to recall is if we have dependent events. And with independent events, the Occurrence of one event doesn't affect the probability of the other event from occurring. So, if we look at the probability of a intersect B, That means that the probability of both events occurring is just going to be equal to the product of the probabilities of each individual event. So that's going to be equal to the probability of event A occurring, multiplied by the probability of event B occurring. So, these are the definitions that we'll be using to determine which of our four choices is correct. So let's start off by looking at choice A, which says if two events are independent, they must be mutually exclusive. And this statement is not correct. And we can provide a counterexample. So consider the event of the first coin landing on my heads and the event of the 2nd coin landing on heads, and these two events are actually independent, but they can occur simultaneously, meaning I could have two heads. So they are also not mutually exclusive. So that is in contrast to their statement that if we have 2 independent events, then they must be mutually exclusive. So that's why choice A is not correct. Now, if you look at choice B, it says that if two events are mutually exclusive, they must be independent. And this here is also not the correct statement. And here we're gonna look at our definitions for mutually exclusive and independence. So, if we look at the probability of A intersect B, we see that for mutually exclusive events, that must be equal to zero, but for uh independent events, that has to be equal to the probability of A multiplied by the probability of B. And so you would actually need 0 to be equal to the probability of a multiplied by the probability of event B. However, this is not true in general, not for any probability of A or any probability of B. It's only true if the probability of A. is equal to 0, or the probability of B is equal to 0. So, since choice B here. Says that they must be independent if they are mutually exclusive. That's not going to be true in general, so that's why it's not correct. But if we look at choice C, for Choice C, it says two events can be both independent and mutually exclusive, if at least one has probability of zero, which is what we see here. That is the conclusion that we have arrived at when we looked at the definitions of mutually exclusive and independence. So, since that statement is true, that means that Cho C is going to be our correct answer. And if you look at choice D for completeness, we'll see that it is also not correct. Since the lack of mutual exclusivity does not guarantee independence, and a counter example for this is to consider drawing two cards without replacement from a deck. The event of the first card being a heart and the event of the second card being a heart are not mutually exclusive. They both could be hearts, but they are not independent because the outcome of the first draw affects the probability of the second draw. If you draw a heart at the first one, there's less hearts left in the decks, so it would have a lower probability than the first draw. So therefore they are not independent events. So that's why choice D is not correct. So the only correct answer for this problem is choice C. Two events can be both independent and mutually exclusive only if at least one has probability zero. So I hope that this explanation has been useful, and I'll see you in the next video.

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True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false,
explain why.
4. When two events are independent, they are also mutually exclusive.

Key Concepts

Independent Events

Mutually Exclusive Events

Relationship Between Independence and Mutual Exclusivity

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True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false,explain why.4. When two events are independent, they are also mutually exclusive.