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 were head. The probability that exactly n elements are chosen is:
-
$\frac{^{2n}\mathrm{C}_n}{4^n}$
-
$\frac{^{2n}\mathrm{C}_n}{2^n}$
-
$\frac{1}{^{2n}\mathrm{C}_n}$
-
$\frac{1}{2}$
A
Correct answer
Explanation
Required favourable cases = Number of ways of selecting n element out of the set of size 2n = $^{2n}\mathrm{C}_n$ As each toss may result in either head or tail, so for 2n tosses, total no. of combinations = 22n = 4n
$\therefore$Required probability = $\dfrac{\text{Favourable cases}}{\text{Total cases}}$
$ = \frac{^{2n}\mathrm{C}_n}{4^n}$