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Binomial: a relation between two probabilities fixes n or p, then find another probability

This type of question, asked 7 times: 2021 Paper 1, 2021 Paper 10, 2021 Paper 25, 14 Jun 2022, 26 Jun 2022, 27 Jun 2022, 10 Apr 2023. All 7 questions below, oldest first.

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  1. Question 1

    2021 Paper 1

    77
    An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :

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  2. Question 2

    2021 Paper 10

    74
    Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to:

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  3. Question 3

    2021 Paper 25

    77
    A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability of getting 2 heads is:

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  4. Question 4

    14 Jun 2022

    77
    If a random variable XX follows the Binomial distribution B(33,p)B(33,p) such that 3P(X=0)=P(X=1)3P(X = 0) = P(X = 1), then the value of P(X=15)P(X=18)−P(X=16)P(X=17)\frac{P(X = 15)}{P(X = 18)}-\frac{P(X = 16)}{P(X = 17)} is equal to

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  5. Question 5

    26 Jun 2022

    77
    Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is:

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  6. Question 6

    27 Jun 2022

    77
    Let XX be a random variable having binomial distribution B(7,p)B(7,p). If P(X=3)=5P(X=4)P(X = 3) = 5P(X = 4), then the sum of the mean and the variance of XX is:

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  7. Question 7

    10 Apr 2023

    158
    Let a die be rolled nn times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is k215\frac{k}{2^{15}}, then kk is equal to:

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