Question

Consider a bettering game where you bet $10 and have a probability of 0.45 of getting $20 back ($10 more than you started with) and a probability of 0.55 of getting no money back (losing the initial $10). The net amount of money gained on each trial is a discrete random variable. Note that losing money can be expressed as a negative gain.

(a) Draw a probability mass function representing this random variable.

(b) Find the expected value of this pmf.

(c) If you start with $50, what is the expected amount of money to be left with after playing 20 times?

Answer 1

(a)

`f(x) = P(X=x) = \begin{Bmatrix} 0.45 \,\,\, \text{for} \,\,\, x = 10 \\ 0.55 \,\,\, \text{for} \,\,\, x = -10 \end{matrix}`

(b)

-1

(c)

30

Explanation:

(a)

Your random variable will have two possible values, 30 and 0, one of them with a probability of 0.45 and the other one with a probability of 0.55. Therefore it looks like this.

`f(x) = P(X=x) = \begin{Bmatrix} 0.45 \,\,\, \text{for} \,\,\, x = 10 \\ 0.55 \,\,\, \text{for} \,\,\, x = -10 \end{matrix}`

(b)

The expected value of this PMF would be

`E[X] = -10*0.55+10*0.45= -1` therefore on average you will have a dollar less.

(c)

For this one, if you play 20 times and your initial amount is 50$ then you have to compute the following expectation.

`E[50+20*X] = 50+20*E[X] = 50-20 = 30`