Question

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.

Answer 1

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Answer 2

Following are the solution to the given question:

Assume three balanced coins were tossed at random, and its sampling area is identified as:

$\bold{S=\{HHH,HHT,HTH,HTT,THH, THT,TTH,TTT\}}$

Let $Y_1$ consider the head number; the choices for $Y_1$ are 0, 1, 2, and 3.

In other words,

$Value \ \ \ \ \ \ \ outcome \\\\0 \ \ \ \ \ \ \ \ \ \ TTT\\\\1 \ \ \ \ \ \ \ \ \ \ HTT,THT,TIH\\\\2 \ \ \ \ \ \ \ \ \ \ HHT,THH, HTH\\\\3 \ \ \ \ \ \ \ \ \ \ HHH$

Let $Y_2$ symbolize the amount of money won on the winning bet in the following way.

• When players get the first heads on the first toss, they gain $\ 1.$
• When you get the first head-on toss 2 or toss 3, you win$\ 2 \ or\ \ 3$, respectively.
• When no heads emerge, you lose $\ 1 (win - \ 1)$.
• $Y_2$  has the following possibilities: $-1,1,2,3$

That is,

$Value \ \ \ \ \ \ \ \ \ \ outcome\\\\-1 \ \ \ \ \ \ \ \ \ \ \ \ TTT\\\\1 \ \ \ \ \ \ \ \ \ \ \ \ HTT, HHT,HTH,HHH\\\\2 \ \ \ \ \ \ \ \ \ \ \ \ \ THT,THH\\\\3 \ \ \ \ \ \ \ \ \ \ \ \ \ TTH$

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