Question

Five thousand lottery tickets are sold for $1 each. One ticket will win $1000, two tickets will win $500 each, and ten tickets will win $100 each. Let X denote the net gain from the purchase of a randomly selected ticket.

a. Construct the probability distribution of C

b. Compute the standard deviation of X

Answer 1

Let X denote the net gain from the purchase of a randomly selected ticket.

For the first prize winner the net gain is 1000-1=999. So, $P(X=999)=\frac{1}{5000}=0.0002$

For second prize winner, $P(X=499)=\frac{2}{5000}=0.0004$

For third prize winner, $P(X=99)=\frac{10}{5000}=0.002$

If you don't win any ticket then $P(X=-1)=1-[P(X=999)+P(X=499)+P(X=99)]$

$=1-[0.0002+0.0004+0.002]=0.9974$

a. Probability distribution

b. Standard deviation

$E(X)=∑X*P(X=x)=999*0.0002+499*0.0004+99*0.002+-1*0.9974=-0.4$

$\sigma=\sqrt{319.64}=17.8785$