If the first head occurs on the first toss you win 1 if, Basic Statistics

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If the first head occurs on the first toss you win 1 if

Three balanced coins are tossed independently. One of the variables of interest is Y1, the number of heads. Let Y2 denote the amount of money won on a side bet in the following manner. If the first head occurs on the first toss, you win $1. If the first head occurs on toss 2 or on toss 3 you win $2 or $3, respectively. If no heads appear, you lose $1 (that is, win -$1).

a Find the joint probability function for Y1 and Y2.

b What is the probability that fewer than three heads will occur and you will win $1 or less?

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