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Flipping a coin we get either head or tail can't get both so events are mutually exclusive. i.e P(A and B) =0. but flipping the same coin twice may result in either head or tail and result of flipping a coin twice is independent of what appeared the first time. So we can say events are independent? if YEs

then how come it is possible that events are mutually exclusive and like wise independent too ?

Junaid Anwar
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  • The words "independent" and "mutually exclusive" apply to different things. The first flip and the second flip of a coin are independent. For a given flip of a coin, the outcomes H and T are mutually exclusive. – jlammy Sep 23 '21 at 17:35

2 Answers2

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If two events $A$ and $B$ are mutually exclusive then $$ A \rightarrow \neg B $$ and $$B \rightarrow \neg A$$ In particular $$P(A\land B) = P(A\cap B) = 0$$ If two events are independent, then $$P(A\land B) = P(A \cap B) = P(A)P(B) $$ If two events are independent and mutually exclusive, then, combining these two results:

$$P(A\cap B) = P(A)P(B) = 0 \iff P(A) = 0 \lor P(B) = 0$$ Therefore, $A$ must have zero probability or $B$ must have zero probability. Note that this does not imply, however, that any of these events are impossible.

econbernardo
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  • Yes mathematically I do agree. But can u please comment on mutually exclusiveness and independence of events if a single coin is tossed twice. Intuitively I feel flipping a coin twice is both mutually exclusive and also independent but mathematically it's not possible so whats ur comment on mutually exclusiveness and independence on flipping a coin twice ? – Junaid Anwar Sep 24 '21 at 02:30
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Let $H_1$ be the event that the first coin was heads. Let $H_2$ be the event the second coin was heads. Similarly, let $T_1$ be the event that the first coin was tails and $T_2$ the event the second coin was tails.

  • $H_1$ and $T_1$ are mutually exclusive.

  • $H_1$ and $T_2$ are independent.

You are confusing the first and second statements with one another. There is no contradiction here. The outcome of the first coin toss is independent of the outcome of the second coin toss, i.e. the outcome of the first flip has no impact on the outcome of the second. The outcome of the first coin toss can not be both possibilities simultaneously, i.e. the event that first coin be heads is mutually exclusive of the event that the first coin is tails.

JMoravitz
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  • Thanks for ur reply. But my confusion still exists. If u can elaborate by considering a single coin flipping twice. Then comment on mutually exclusive events and independent event. – Junaid Anwar Sep 24 '21 at 02:28
  • @JunaidAnwar Let $H_1$ be the event that the first time the coin was flipped it was a heads... and $H_2$ be the event that the second time the coin was flipped the coin being heads. Similarly adjust the wording for $T_1,T_2$. It doesn't matter that it was the same coin versus two different coins in the slightest. – JMoravitz Sep 24 '21 at 03:14
  • @JunaidAnwar when referring to "heads is mutually exclusive from tails" you are not being rigorous enough in the statement and you need to be more specific as to what "heads" is in terms of outcomes of specific flips. "Heads for the first flip is mutually exclusive from tails also for the first flip" is true. "Heads for the first flip is not mutually exclusive and is instead independent from tails for the second flip (noting the first flip is a different flip than the second flip)" – JMoravitz Sep 24 '21 at 03:20
  • well, now we are getting a bit closer. Let me be rigorous. The Head of the first flip is mutually exclusive from the tail of the first flip. now what I am curious about is Head of the first flip is either independent or dependent from the tail of the first flip?? because mathematically once events get mutually exclusive these events can't be independent unless P (one of them)=0. – Junaid Anwar Sep 24 '21 at 19:58
  • $H_1$ is dependent on $T_1$ – JMoravitz Sep 24 '21 at 20:06
  • Thanks a lot I am clear now. – Junaid Anwar Sep 25 '21 at 07:33