For each element in a set of size 2n, an unbiased coin is tossed. The 2n coin…
2006
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 is a Head. The probability that exactly n elements are chosen is:
- A.
C(2n, n) / 4^n
- B.
C(2n, n) / 2^n
- C.
1 / C(2n, n)
- D.
1/2
Attempted by 2 students.
Show answer & explanation
Correct answer: A
Each of the 2n elements is independently chosen with probability 1/2, because an unbiased coin is tossed for each element and the element is chosen on Head.
Therefore, the number of chosen elements follows a binomial distribution with parameters 2n and 1/2.
The probability that exactly n elements are chosen is:
C(2n, n) (1/2)^n (1/2)^(2n - n)
= C(2n, n) (1/2)^(2n)
= C(2n, n) / 2^(2n)
= C(2n, n) / 4^n.
Therefore, the correct probability is C(2n, n) / 4^n.