If I toss a coin 3 times and want to know the probability of at least one head, I have understood th

varitero5w 2022-06-26 Answered
If I toss a coin 3 times and want to know the probability of at least one head, I have understood that the answer is 1 0.5 3 = 99 %. However, why cannot I not use the additon rule P ( A B ) = P ( A ) + P ( B ) P ( A B ), i.e. 0.5 + 0.5 + 0.5 0.5 3 ?
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Answers (2)

Angelo Murray
Answered 2022-06-27 Author has 23 answers
I assume you are talking about the Inclusion-exclusion principle when you say addition rule.
You cannot used the addition rule for this problem because you are discussing 3 coins in your problem. If you were discussing two, the addition rule above would be enough.
The correct formula for three coins would be P ( A B C ) = P ( A ) + P ( B ) + P ( C ) P ( A B ) P ( B C ) P ( C A ) + P ( A B C )The answers is, therefore 0.5 + 0.5 + 0.5 0.5 2 0.5 2 0.5 2 + 0.5 3 = 0.875 = 1 0.5 3
However, in problems like this note that it is better not to use the addition rule.
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Jaqueline Kirby
Answered 2022-06-28 Author has 6 answers
Make sure you are careful when you define and count your events. For example, for this problem, we can say A is the event of getting EXACTLY ONE HEAD. Let B be the event of getting EXACTLY TWO HEADS. Let C be the event of getting EXACTLY THREE HEADS. Then the probability you are looking for is P ( A ) + P ( B ) + P ( C ). Notice we don't subtract anything because the events are distinct (no overlaps). P ( A ) = 3 ( 1 / 2 ) 3 = 3 / 8, P ( B ) = 3 ( 1 / 2 ) 3 = 3 / 8, and P ( C ) = ( 1 / 2 ) 3 = 1 / 8. The sum is 7/8, which is the same as the other way, which gave 1 ( 1 / 2 ) 3 = 7 / 8.
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