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Repeat a trial until the first success (geometric probabilities, alternate throws)

This type of question, asked 5 times: 2021 Paper 19, 11 Apr 2023, 27 Jan 2024, 29 Jan 2024, 24 Jan 2025. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 19

    69
    A fair die is tossed until six is obtained on it. Let XX be the number of required tosses, then the conditional probability P(X≥5∣X>2)P(X \geq 5 \mid X > 2) is:

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  2. Question 2

    11 Apr 2023

    172
    Let the probability of getting head for a biased coin be 14\frac{1}{4}. It is tossed repeatedly until a head appears. Let NN be the number of tosses required. If the probability that the equation 64x2+5Nx+1=064x^{2}+ 5Nx + 1 = 0 has no real root is pq\frac{p}{q}, where pp and qq are co-prime, then q−pq - p is equal to

  3. Question 3

    27 Jan 2024

    86
    A fair die is tossed repeatedly until a six is obtained. Let XX denote the number of tosses required and let a=P(X=3),b=P(X≥3)a = P(X = 3),b = P(X \geq 3) and c=P(X≥6∣X>3)c = P(X \geq 6 \mid X > 3). Then b+ca\frac{b + c}{a} is equal to

  4. Question 4

    29 Jan 2024

    64
    A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is

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  5. Question 5

    24 Jan 2025

    61
    A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8 , and BB wins if he throws a sum of 8 before AA throws a sum of 5 . The probability, that A wins if A makes the first throw, is

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