The textbook states that mutually exclusive events are dependent because one situation can't happen without the other. Why? I just can't seem to wrap my head around it. Why can't mutually exclusive events be independent. Can you please provide an example along with an explanation.
Thank you. It's greatly appreciated.
Thank you. It's greatly appreciated.
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In statistics (as in mathematics in general) always go back to basic definitions.
What is the definition of independent events? The probability of one event does not affect the probability of another event.
I have a fair coin (50% change of heads, 50% chance of tail).
The coin has no memory.
I toss the coin. Head.
What is the probability that the next toss will be head? 50%
OK, I have now completed the two tosses.
I pretend to slowly walk away, then I suddenly turn around, grab the coin and toss it high in the air. What is the probability that it now lands "head"?
50%
this toss is independent from the previous tosses. It is also independent from me faking departure and suddenly turning around.
I could drink a coffee, it changes nothing to the probability of the coin toss. Independent events.
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I have two balls in a bag. One red, one blue.
What is the probability that I pick the red on the first draw? 50%.
I do NOT replace the red ball in the bag.
What is the probability that I pick the red ball on the second draw?
it is NOT 50%. Therefore, it is not an independent event.
As it happens in this case, this is a mutually exclusive event. The fact that I picked red on the first draw EXCLUDES the probability of me picking red again on the second draw. This makes the probability 0% on the second draw.
For the definition, the only important item of information is that the probability, on the second draw, is NOT the same as in the first draw. The fact that it becomes 0% just makes it an extreme case of dependence.
What is the definition of independent events? The probability of one event does not affect the probability of another event.
I have a fair coin (50% change of heads, 50% chance of tail).
The coin has no memory.
I toss the coin. Head.
What is the probability that the next toss will be head? 50%
OK, I have now completed the two tosses.
I pretend to slowly walk away, then I suddenly turn around, grab the coin and toss it high in the air. What is the probability that it now lands "head"?
50%
this toss is independent from the previous tosses. It is also independent from me faking departure and suddenly turning around.
I could drink a coffee, it changes nothing to the probability of the coin toss. Independent events.
---
I have two balls in a bag. One red, one blue.
What is the probability that I pick the red on the first draw? 50%.
I do NOT replace the red ball in the bag.
What is the probability that I pick the red ball on the second draw?
it is NOT 50%. Therefore, it is not an independent event.
As it happens in this case, this is a mutually exclusive event. The fact that I picked red on the first draw EXCLUDES the probability of me picking red again on the second draw. This makes the probability 0% on the second draw.
For the definition, the only important item of information is that the probability, on the second draw, is NOT the same as in the first draw. The fact that it becomes 0% just makes it an extreme case of dependence.
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