Question

The chance of winning a game is 1 in approximately 7 million. Suppose you buy a $20 ticket in anticipation of winning the $4 million grand prize. Let X denote the winning. What is the expected winning for this game? Interpret the result. Please type your answer.

Answer 1

TOPIC:Expected value of random variables.

Let us define, the random variable X as-

X = The net winnings of the game ( in $).

Now, consider these following two cases as--

CASE- I (You won the grand prize) :

Here,

Ticket price (T) = $ 20.

Winning amount (W) = $ 4,000,000.

The net winnings (X) = (W - T) = $ (4,000,000 - 20)= $ 3,999,980.

Corresponding probability (P(X))

= P( of winning the game).

( Since, the chance of winning the game is 1 in approximately 7 million.)

CASE -2 (You did not win the grand prize):

Here,

Ticket price (T) = $20.

Winning amount (W) = $0.

The net winnings (X) = (W-T) = $ -20.

Corresponding probability (P(X)) :

=P( of not winning the game).

= 1 - P( of winning the game).

Thus, here the discrete random variable X takes the values 3,999,980 and -20, along with the probabilities

1/ 7,000,000 and 6,999,999/7,000,000 respectively.

We need to find,

The expected net winnings for the this game.

(Rounded to two decimals.)

Hence, the expected net winnings for this game is = $ - 19.43.

INTERPRETATION OF THE RESULT:

A person who plays the game for a single time, can expect a loss of $19.43.