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For each element in a set of size $2n$, an unbiased coin is tossed. The $2n$ coin tosses are independent. An element is chosen if the corresponding coin toss was a head. The probability that exactly $n$ elements are chosen is

  1. $\frac{^{2n}\mathrm{C}_n}{4^n}$
  2. $\frac{^{2n}\mathrm{C}_n}{2^n}$
  3. $\frac{1}{^{2n}\mathrm{C}_n}$
  4. $\frac{1}{2}$

7 Answers

Best answer
40 votes
40 votes

Answer - A

Ways of getting $n$ heads out of $2n$ tries $= ^{2n}C_{n}.$

Probability of getting exactly $n$-heads and $n$-tails $= \left(\dfrac{1}{2^n}\right).\left(\dfrac{1}{2^n}\right)$

Number of ways $= \dfrac{^{2n}C_{n}}{4^n}.$

edited by
28 votes
28 votes

Required Probability=$\frac{No. of Favourable Ways }{Total no. of Ways}$

No. of favourable ways= 2nC (bcoz select exactly n heads out of 2n tosses )

Total no. of ways= 2n tosses and each have 2 possibilities either H or T so total=$2^{2n}$=$4^n$ possibilities

So Ans is $\frac{\binom{2n}{n} }{4^{n}}$  which fits option A.

edited by
2 votes
2 votes
The question is mainly about probability of n heads out of 2n coin tosses.
P = 2nCn∗((1/2)^n)∗((1/2)^n) = (2nCn) / (4^n)
Answer:

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