S0 = 0 and EXERCISE 1.7.– Consider the following game of chance. A player begins by choosing a number between 6 and 8 (inclusive), which we call the principal. The player then rolls 2 uncut, six-sided, non-rigged dice and sums the result. The wins are as follows:
– If the sum is 2 or 3, the player loses 1 DT (Tunisian dinar).
– If the sum is 11, the player wins 1 DT if the principal is 7; otherwise, they lose 1 DT.
– If the sum is 12, the player wins 1 DT if the principal is 6 or 8; otherwise, they lose 1 DT.
– Finally, in all other cases, nothing happens (no win, no loss).
1 1) Determine Ω, the universe of all outcomes of the experiment.
2 2) S is the random variable giving the sum of the two dice. Determine the distribution of S.
3 3) X7 is the random variable giving the player’s winnings (negative in the case of a loss) when the principal is 7. Determine the distribution of X7.
4 4) Calculate [X7] and (X7).
5 5) We consider X6 to be the winnings of the player when the principal is 6. Determine the distribution of X6 and then calculate [X6].
EXERCISE 1.8.– Let p ∈]0, 1[. We have one coin that leads to tails with the probability p. We toss this coin until we obtain tails for the second time. Let X be the number of heads obtained during this experiment.
1 1) Determine the distribution of X.
2 2) Justify the existence of the expectation of X.
3 3) Calculate [X].
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