A) \[\frac{^{2n}{{C}_{n}}}{{{2}^{2n}}}\]
B) \[\frac{^{2n}{{C}_{n-1}}}{{{2}^{n}}}\]
C) \[\frac{n}{{{2}^{2n}}}\]
D) None
Correct Answer: A
Solution :
[a] The number of possible outcomes of 2n tosses is \[{{2}^{2n}}.\] There are \[^{n}{{C}_{r}}\]ways of getting r heads, with \[0\le r\le n,\] in n tosses. Therefore, the number of ways of getting r heads in both the first n and last n tosses is \[{{{{(}^{n}}{{C}_{r}})}^{2}}.\]Summing over all values of r. the number of favourable ways is \[{{{{(}^{n}}{{C}_{0}})}^{2}}+{{{{(}^{n}}{{C}_{1}})}^{2}}+{{{{(}^{n}}{{C}_{2}})}^{2}}+...+{{{{(}^{n}}{{C}_{n}})}^{2}}{{=}^{2n}}{{C}_{n}},\] So that the required probability is \[\frac{^{2n}{{C}_{n}}}{{{2}^{2n}}}.\]
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