If events E and F are disjoint and the events F and G are disj...

Question

If events E and F are disjoint and the events F and G are disjoint, must the events E and G necessarily be disjoint? Give an example to illustrate your opinion.

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Let D and E be disjoint events and E and F also be disjoint. Must D and F be disjoint? Use an example to justify your answer. So it appears for this particular problem, we are told that D and E are disjoint events and ENF are also disjoint events, and we're asked to determine must DNF be disjoint. And we're asked to use an example to justify our answer. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer may be. A is yes, D and F must be disjoint. B is no. D and F do not have to be disjoint. C is only if D, E, and F are all singletons, and D is only if D and F are complements. OK. So our first step that we need to take is we need to recall a note based on our prior knowledge that disjoint events have no outcomes in common. So, as a result, we could safely write that the intersection of set D and E is equal to the empty set, which the empty set symbol is a 0 or an O with a diagonal line going through it. And the intersection symbol is an upside down union symbol. So once again, to quickly recap, the intersection of set D and E is equal to the empty set. And The intersection of set E and set F is also equal to the empty set. OK. So once again, based on our prior knowledge, disjoint events have no outcomes in common. So for our second step, we need to note that the question asks if the intersection of set D and set F is equal to the empty set must also be true. So with that in mind, our third step is we need to note that we need to let the sample space B. S is equal to the curly brackets of A, B, C. With once again with D being equal to curly brackets of A. And E being equal to curly brackets of B. And F being equal to the curly brackets of drum roll, A C. Fantastic. So, once again, Moving right along here, our 4th step will state that the intersection of set D and set E Is equal to curly brackets. Of so once again. The intersection of set D and set E is equal to the intersection of curly brackets of A. So once again, the set of curly brackets of A. And the set curly brackets of B. Is equal to the empty set. So that will mean as a result that D and E are disjoint. And our 5th step is we need to note that the intersection of set E and set F is equal to the intersection of the curly brackets of B and the set of curly brackets. Of a C. is equal to the empty set. So that will mean as a result that E and F are also disjoint. So step number 6 is we need to note that the intersection of set D and set F is equal to the intersection of set curly brackets of A. And the set of curly brackets of A, C, which will be equal to curly brackets of A, which is not equal, it cannot equal the empty set. So curly brackets of A cannot equal the empty set. So that will mean based on this piece of information that D and F are not disjoint. So, That will mean for our 7th and final step that therefore as a result, D and F do not have to be disjoint even if D and E and E and F are considered disjoint. So as a result, our final answer without a doubt will be multiple choice answer letter B, which once again is no, D and F do not have to be disjoint. And that's it, we've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

If events E and F are disjoint and the events F and G are disjoint, must the events E and G necessarily be disjoint? Give an example to illustrate your opinion.

Key Concepts

Disjoint (Mutually Exclusive) Events

Transitivity of Disjointness

Counterexample in Probability

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