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A fair coin with sides marked heads and tails is to be [#permalink]

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08 Dec 2007, 09:36

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A fair coin with sides marked heads and tails is to be tossed eight times. What is the probability that the coin will land tails side up more than five times?

A fair coin with sides marked heads and tails is to be tossed eight times. What is the probability that the coin will land tails side up more than five times?

a) 37/256

b) 56/256

c) 65/256

d) 70/256

e) 81/256

Please provide your explanation

This actually is not that bad at all. I did it in less than 2 min in my head. Just gotta think about it logically.

List the winning scenarios: 6 tails 2 heads 1/2^8 we have 8!/6!2! ways to arrage TTTTTTHH so 28/256 7 tails 1 head 8!/1!7! ---> 8/256

I tried to go from the other side and made a mistake.

the wrong logic: the probability more than 4 tails is \(\frac12\) (due to symmetry), the probability for 5 tails is \(\frac{C^8_5}{2^8}=\frac{56}{256}\). therefore, \(p=\frac12-\frac{56}{256}=\frac{72}{256}\)

the right logic: we have 9 variants for the number of tails: 0,1,2,3,4,5,6,7,8. Therefore, correct formula with symmetry approach is: \(p=\frac12-\frac12*\frac{C^8_4}{2^8}-\frac{C^8_5}{2^8}=\frac{128}{256}-\frac12*\frac{70}{256}-\frac{56}{256}=\frac{37}{256}\)
_________________

A fair coin with sides marked heads and tails is to be tossed eight times. What is the probability that the coin will land tails side up more than five times?

a) 37/256

b) 56/256

c) 65/256

d) 70/256

e) 81/256

Please provide your explanation

First mistake which I did, I calculated it as "at least 5 times", but question clearly states "more" than five times! I should be more careful in reading the questions!

A fair coin with sides marked heads and tails is to be tossed eight times. What is the probability that the coin will land tails side up more than five times?

a) 37/256

b) 56/256

c) 65/256

d) 70/256

e) 81/256

Please provide your explanation

First mistake which I did, I calculated it as "at least 5 times", but question clearly states "more" than five times! I should be more careful in reading the questions!

How my mistake the same to your! I start off by 5 coins landing tail. And one more, I calculate 2^8 is only 64. Comparing with Answer series, my god, I made a stupid mistake.

A fair coin with sides marked heads and tails is to be tossed eight times. What is the probability that the coin will land tails side up more than five times?

A fair coin with sides marked heads and tails is to be tossed eight times. What is the probability that the coin will land tails side up more than five times?

a) 37/256

b) 56/256

c) 65/256

d) 70/256

e) 81/256

Soln: For more than 5 times it can be broken into = Prob(that tails appears 6 times) + Prob(that tails appears 7 times) + Prob(that tails appears 8 times) using formula nCr * p^r * q^(n-r) = 8C6 * (1/2)^6 * (1/2)^2 + 8C7 * (1/2)^7 * (1/2)^1 + 8C8 * (1/2)^8 * (1/2)^0 = 37/(2^8)

Re: A fair coin with sides marked heads and tails is to be [#permalink]

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09 Dec 2013, 02:57

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A fair coin with sides marked heads and tails is to be tossed eight times. What is the probability that the coin will land tails side up more than five times? A. 37/256 B. 56/256 C. 65/256 D. 70/256 E. 81/256

The probability that the coin will land tails side up more than five times equals to the sum of the probabilities that coin lands 6, 7 or 8 times tails side up.

\(P(t=6)=\frac{8!}{6!2!}*(\frac{1}{2})^8=\frac{28}{256}\), we are multiplying by \(\frac{8!}{6!2!}\) as the scenario tttttthh can occur in \(\frac{8!}{6!2!}=28\) # of ways (tttttthh, ttttthth, tttthtth, ... the # of permutations of 8 letters tttttthh out of which 6 t's and 2h's are identical);

\(P(t=7)=\frac{8!}{7!}*(\frac{1}{2})^8=\frac{8}{256}\), the same reason of multiplication by \(\frac{8!}{7!}=8\);

\(P(t=8)=(\frac{1}{2})^8=\frac{1}{256}\), no multiplication as the scenario tttttttt can occur only in one way;

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