Question

In a game of one-spot Keno, a card is purchased for $2.00.

It allows a player to choose one number from 1 to 80. The
dealer then chooses twenty numbers at random from 1 to
80. If the player's number is among the numbers chosen
by the dealer then the player is paid $3.30, but does not
get to keep the $2.00 paid to play the game.
What is the expected value of buying one card?
The expected value of buying one card is______
Hint

Answer 1

Explanation:

The expected value of a random variable is the sum of the products of all possible outcomes and their respective probabilities. In this case, the random variable is the player's payout, which can be $3.30 if the player's number is chosen by the dealer, or $0 if it is not.

Let p be the probability that the player's number is chosen by the dealer. Since there are 80 possible numbers, and the dealer chooses 20 numbers at random, the probability that any one number is chosen is 20/80, or 1/4. Therefore, the probability that the player's number is chosen is also 1/4, or 0.25.

The expected payout can be calculated as follows:

Expected payout = (Payout if number is chosen) x Probability of number being chosen + (Payout if number is not chosen) x Probability of number not being chosen

Expected payout = ($3.30) x (0.25) + ($0) x (0.75)

Expected payout = $0.825

Since the player paid $2 to play the game, the net expected payout is:

Net expected payout = Expected payout - Cost to play the game

Net expected payout = $0.825 - $2.00

Net expected payout = -$1.175

Therefore, the expected value of buying one card is -$1.175, which means that on average, the player can expect to lose $1.175 for every card purchased. This is a negative expected value, indicating that the game is not a good bet for the player.