Statistics And Probability Question 75
Question: One hundred identical coins, each with probability P of showing up heads, are tossed. If 0<p<1 and the probability of heads showing on 50 cons is equal to that of heads showing on 51 cons. The value of p is
Options:
A) $ \frac{1}{2} $
B) $ \frac{49}{101} $
C) $ \frac{50}{101} $
D) $ \frac{51}{101} $
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Answer:
Correct Answer: D
Solution:
- [d] Let $ X\tilde{\ }B(100,p) $ be the number of coins showing heads, and let $ q=1-p. $ Then, since $ P(X=51)=P(X=50), $ we have $ ^{100}C_{51}(P^{51})(q^{49}){{=}^{100}}C_{50}(p^{50})(q^{50}) $
$ \Rightarrow \frac{p}{q}=( \frac{100!}{50!50!} )( \frac{51!49!}{100!} ) $
$ \Rightarrow \frac{p}{1-p}=\frac{51}{50}\Rightarrow 50p=51-51p\Rightarrow p=\frac{51}{101} $
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