Ask Question
2 December, 17:05

Two fair dice are rolled. Find the joint probability mass function of X and Y when (a) X is the largest value obtained on any die and Y is the sum of the values; (b) X is the value on the first die and Y is the larger of the two values; (c) X is the smallest and Y is the largest value obtained on the dice.

+4
Answers (1)
  1. 2 December, 17:24
    0
    a)

    P (X = x₀, Y = 2x₀) = 1/36

    P (X = x₀, Y = k) = 1/18 for k between x₀+1 and 2x₀-1 inclusive

    Every other event has probability 0. x₀ is any number between 1 and 6 inclusive.

    b)

    P (X = x₀, Y = x₀) = x₀/36

    P (X = x₀, Y = k) = 1/36 for k between x₀+1 and 6 inclusive.

    x₀ is between 1 and 6 inclusive. Every other event has probability 0.

    c)

    P (X = x₀, Y = x₀) = 1/36

    P (X = x₀, Y = k) = 1/18 with k between x₀+1 and 6 inclusive

    x₀ between 1 and 6 inclusive. Any other event has probability 0.

    Step-by-step explanation:

    Note that there are 36 possible results for the dice

    a)

    P (X = 1, Y = 2)

    This is obtained only when both dices are 1, hence its probability is 1/36

    P (X = 1, Y = k) = 0 (k > 1)

    because if the largest value of the dice is 1, then both dices are 1

    P (X = 2, Y = 3)

    one dice is 2, the other one is 3, hence there are 2 possibilities and the probability is 2/36 = 1/18

    P (X = 2, Y = 4)

    This happens only if both dices are 2, hence the probability is 1/36.

    P (X = 2, Y = k) = 0 (k > 2)

    same argument of above. If the largest dice is 2, then the sum is either 3 or 4.

    P (X = 3, Y = 4), P (X = 3, Y = 5)

    in both given events we need one dice to be 3 and the other dice to be 1 for the first event and 2 for the second event. In both cases, there are only 2 favourable cases, hence the probability of the event is 2/36 = 1/18

    P (X = 3, Y = 6)

    This event happens only when both dices are 3, hence the probability is 1/36

    This should show a pattern. As long as x₀ is between 1 and 6, if y₀ is between x+1 and 2x-1, then the probability P (X = x₀, Y = y₀) is 1/18 (either first dice is x₀, second dice is y₀-x₀ or first dice is x₀ and second dice is y₀ - x₀), also P (X = x₀, Y = 2x₀) = 1/36 (both dices are x₀). Every other event has probability 0.

    b) We can separate them using conditional probability and the fact that both dices results are independent with each other.

    P (X = x₀, Y = y₀) = P (X = x₀) * P (Y = y₀ | X = x₀)

    P (X = x₀) = 1/6 for any value x₀ between 1 and 6.

    If y₀ is x₀, this means that the first dice has the largest value, so the second dice is between 1 and x₀, and the probability of this event is x₀/6 (x₀ favourable cases over 6 possible ones).

    If y₀ is not x₀, then it should be higher (otherwise the event would be impossible and it would have probability 0). As long as y₀ is between 2 and 6, the probability of this event is 1/6.

    Thus

    P (X = x₀, Y = x₀) = 1/6 * x₀/6 = x₀/36

    P (X = x₀, Y = x₀ + k) = (1/6) ² = 1/36 (k > 0)

    Every other probability is 0

    c)

    P (X = x₀, Y = x₀) = 1/36 (because both dices are equal to x₀ in this event)

    P (X = x₀, Y = x₀+k) = 2/36 = 1/18 (here k > 0. One possibility is the first dice is x₀ and the second one is x₀+k, and the remaining possibility is the first dice is x₀+k and the second dice is x₀)

    Evert other event has probability 0.
Know the Answer?
Not Sure About the Answer?
Find an answer to your question ✅ “Two fair dice are rolled. Find the joint probability mass function of X and Y when (a) X is the largest value obtained on any die and Y is ...” in 📘 Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions.
Search for Other Answers