GATE2006-21

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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. $\frac{1}{2}$
5

As here they are talking about choosing n elementsout of 2n elements. so they can choose 1 element out of 2n or 2 elements out of 2n.....................2n elements out of 2n.

so total no of cases 2nc1 + 2nc2 + 2nc3+.............+2nc2n= 2^(2n)= 4^n

ways of getting n heads out of 2n tries = 2nCn

probability of getting exactly n heads and n tails = (1/2n)(1/2n)

number of ways = 2nCn/4n

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
0
what if we do as below:
no. of ways to get n heads = 2nCn
no. of total outcome = 2n
therefore, probability of getting exactly n heads out of 2n tosses = 2nCn /2n

what does it calculate?
1
How is it saying n heads out of 2n?
3
We can simply solve this using Bionomial Distribution

Here is the link for theory- http://stattrek.com/probability-distributions/binomial.aspx

1
the best way to solve...

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
1 vote
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)

We can also put small values for n and eliminate options.

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