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Binomial probability of exactly or at least k successes with given n and p

This type of question, asked 5 times: 2021 Paper 17, 6 Apr 2023, 31 Jan 2024, 4 Apr 2024, 24 Jan 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 17

    66
    A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. The smallest value of nn, so that the probability of guessing at least ' nn ' correct answers is less than 12\frac{1}{2}, is:

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  2. Question 2

    6 Apr 2023

    62
    A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is k311\frac{k}{3^{11}}, then kk is equal to

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  3. Question 3

    31 Jan 2024

    168
    A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is-

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  4. Question 4

    4 Apr 2024

    177
    In a tournament, a team plays 10 matches with probabilities of winning and losing each match as 13\frac{1}{3} and 23\frac{2}{3} respectively. Let xx be the number of matches that the team wins, and yy be the number of matches that team loses. If the probability P(∣x−y∣≤2)P(|x - y| \leq 2) is pp, then 39p3^{9}p equals
  5. Question 5

    24 Jan 2026 S1

    67
    From a lot containing 10 defective and 90 nondefective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is :-

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