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Fair coins mixed with one two-headed coin: probability of a head, or that the coin was fair

This type of question, asked 2 times: 7 Apr 2025, 6 Apr 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    7 Apr 2025

    130
    A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is mn,gcd(m,n)=1\frac{m}{n},gcd(m,n) = 1, then n2−m2n^{2}-m^{2} is equal to:

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  2. Question 2

    6 Apr 2026 S1

    60
    A bag contains (N+1)(N + 1) coins -N fair coins, and one coin with 'Head' on both sides. A coin is selected at random and tossed. If the probability of getting 'Head' is 916\frac{9}{16}, then N is equal to:

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