Give a complete explanation for your answers to the following: a) Let A and B be

Question

Give a complete explanation for your answers to the following: a) Let A and B be mutually exclusive events. Are A and B independent or dependent? Or could there be cases of mutually exclusive events that are independent and cases that are dependent? If there are cases of each, give examples of independent MR events and of dependent MF, events. b) Now let A and B be collectively exhaustive events (but not necessarily mutually exclusive). Are A and B independent or dependent0 Or could there be cases of collectively exhaustive events (hat are independent and cases that are dependent? If there arc cases of each, give examples of independent CE events and of dependent CE events.

Explanation / Answer

Solution :-

a) Mutually exclusive events are almost never independent. Lets discuss the same with some examples:-

Consider a fair coin and a fair six-sided die. Let event A be obtaining heads, and event B be rolling a 6. Then we can reasonably assume that events A and B are independent, because the outcome of one does not affect the outcome of the other. The probability that both A and B occur is

Pr[A and B] = Pr[A]Pr[B] = (1/2)(1/6) = 1/12.

Since this value is not zero, then events A and B cannot be mutually exclusive.

An example of a mutually exclusive event is the following: Consider a fair six-sided die as before, only in addition to the numbers 1 through 6 on each face, we have the property that the even-numbered faces are colored red, and the odd-numbered faces are colored green.
Let event A be rolling a green face, and event B be rolling a 6. Then

Pr[A] = 1/2
Pr[B] = 1/6

as in our previous example. But it is obvious that events A and B cannot simultaneously occur, since rolling a 6 means the face is red, and rolling a green face means the number showing is odd. Therefore,

Pr[A and B] = 0.

Therefore, we see that a mutually exclusive pair of nontrivial events are also necessarily dependent events. This makes sense because if A and B are mutually exclusive, then if A occurs, then B cannot occur; and vice versa. This stands in contrast to saying the outcome of A does not affect the outcome of B, which is independence of events.

b)

"Mutually exclusive" means of any two possible outcomes A and B, the logical expression A and B cannot occur. Put another way, two propositions are mutually exclusive if the following are true: "If A, then not B" and "If B, then not A."

"Collectively exhaustive" means of any set of outcomes A, B, C, D, … Z, at least one of the outcomes must occur. For instance, if you roll a die, you will get a result that is "1", "2", "3", "4", "5", or "6". You cannot get a "7", or a "3½" or a "", nor can you get no result at all. You are limited to "1" through "6", which are thus said to be collectively exhaustive.

"Independent" is an outcome the probability of which does not affect the probability of another outcome.

For an example of a type of probability that involves both these principles, take flipping a coin. The coin will come up heads or tails. It cannot be both, ("heads" and "tails" are mutually exclusive) but neither can it be something different from "heads" or "tails" ("heads" and "tails" are collectively exhaustive).

However, if you flip the coin twice, and you take the two results separately, you will find that the second coin toss is independent from the first—it's entirely possible that the coin could come up heads twice, tails twice, heads the first time and tails the second time, or tails the first time and heads the second time. However, the range of outcomes is still collectively exhausted by "heads" and "tails".

Now consider an example of flipping a coin and rolling a die, outcomes will be {H,T} and {1,2,3,4,5,6}, and the event is getting Tail followed by 6 on the die. Now the outcomes are dependent and as it is one of the possible outcome of the sample, it is colloectively exhaustive as well.

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