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Dice with unusual faces or face probabilities: probability of a given total

This type of question, asked 4 times: 2021 Paper 7, 2021 Paper 20, 25 Jun 2022, 23 Jan 2025. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 7

    69
    Two dices are rolled. If both dices have six faces numbered 1,2,3,5,7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is :

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  2. Question 2

    2021 Paper 20

    77
    When a certain biased die is rolled, a particular face occurs with probability 16−x\frac{1}{6}- x and its opposite face occurs with probability 16+x\frac{1}{6}+x. All other faces occur with probability 16\frac{1}{6}. Note that opposite faces sum to 7 in any die. If 0<x<160 <x<\frac{1}{6}, and the probability of obtaining total sum =7= 7, when such a die is rolled twice, is 1396\frac{13}{96}, then the value of xx is:

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  3. Question 3

    25 Jun 2022

    166
    A biased die is marked with numbers 2,4,8,162,4,8,16, 32,32 on its faces and the probability of getting a face with mark nn is 1n\frac{1}{n}. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48 , is

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  4. Question 4

    23 Jan 2025

    70
    One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is

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