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Fair coin binomial: probabilities equal to 1/2 by symmetry

This type of question, asked 4 times: 11 May 2023 S1, 23 Apr 2025 S1, 11 Apr 2026 S2, 18 Apr 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    11 May 2023 S1

    109
    The p.m.f. of a random variable X is given by P[X=x]=(5x)125P[X=x] = \binom{5}{x}\frac{1}{2^5} if x=0,1,2,3,4,5x = 0,1,2,3,4,5; 0 otherwise. Then which of the following is not correct?

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  2. Question 2

    23 Apr 2025 S1

    101
    A fair coin is tossed 100 times. The chance of getting a head even number of times is

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  3. Question 3

    11 Apr 2026 S2

    99
    The probability mass function of a random variable X is given by P(X=x)=5Cx25P(X = x) = \dfrac{{}^5C_x}{2^5}, for x=0,1,2,3,4,5x = 0, 1, 2, 3, 4, 5 and =0= 0, otherwise. Then which of the following is correct?

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  4. Question 4

    18 Apr 2026 S1

    98
    A fair coin is tossed 9 times. On each toss, a man predicts that the outcome will be heads. The probability that the number of successful predictions is strictly greater than the number of unsuccessful predictions is...

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