Multiple choice

A certain coin is tossed with probability of showing head being p. let q denotes the probability that when the coin is tossed four times the number of heads obtained is even then

  1. 2q= 1 +(2p-1)4

  2. q = l + (2p + 1 ) 4

  3. q = 1 - (2p - 1 )4

  4. 2q = 1 - (2p + 1 )4
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A Correct answer
Explanation

The probability of an even number of heads in n trials is given by (1 + (1-2p)^n)/2. For n=4, this is (1 + (1-2p)^4)/2. Since (1-2p)^4 = (2p-1)^4, the expression becomes (1 + (2p-1)^4)/2, or 2q = 1 + (2p-1)^4.

AI explanation

By the binomial probability formula, the probability of getting an even number of heads (0, 2, or 4) is q = (p+q)^4 + (p-q)^4 over 2, where q represents the probability of tails in the formula, not to be confused with the variable q in the question. Substituting the tail probability as (1-p), the sum of an even number of heads is (1 + (2p-1)^4) divided by 2. Multiplying both sides by 2 gives the identity 2q = 1 + (2p-1)^4.

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