Dattaprabhakar G. answered • 09/20/14
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Rick:
People often forget to mention THREE crucial assumptions and, unfortunately, answer the question anyway.
Assumption 1: All the tosses are independent. For example, if the first person's first toss is Heads, he does NOT use a two-headed coin, then on!
Assumption 2: The probabilities P(Head) and P(Tail) remain the same from toss to toss.
Assumption 3: The coins are "fair"; P(H) = 0.5, P(T)= 0.5. THIS assumption is NOT necessary to get a more realistic answer. Let me show you. let P(H) = p and P(T) = 1 - p.
Then, under Assumptions ! and 2, the outcomes of interest and their probabilities are:
T T T T (1-p)4
T H T H (1-p) p(1-p)p = p2(1-p)2 ,
T H H T p2(1-p)2
H T H T p2(1-p)2
H T T H p2(1-p)2
So the more realistic answer to your question "the probability that they get the same number of tails" is
(1-p)4 + 4 p2(1-p)2.
Obviously, when p = 0.5, the answer is 5/16. But that need not be always the case. For example, if p = 0.9, (coin loaded in favor of heads) then the answer is (0.1)4 + 4 (0.9)2(0.1)2. Please compute what it is (definitely NOT 5/16) and post a comment. Thanks.
Dr. G.
Nadia N.
Dr. G,
I DID assume that the coins are "fair". I also assumed that Michael is not taking Stats and Probabilities, but Algebra. Both my assumptions might be wrong.
That being said, yes, your answer is the right one :)
Nadia
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09/20/14
Dattaprabhakar G.
Michael F.
You are absolutely right! Thanks for pointing that out. I was using Nadia N.'s answer, but should have given more thought. The correct answer, under Assumptions 1 and 2, adds the prob of H H H H , p4, to the old answer.
The correct answer is p4 + (1-p)4 + 4 p2(1-p)2.
If the coins are further assumed to be fair, this answer is 6/16 = 0.375. I had asked Rick to compute the old answer for p= 0.9. I will compute the correct answer here. It is 0.6886. Far cry from 0.375!!!
Dr. G.
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09/20/14
Michael F.
09/20/14