Provide an appropriate response.
In a game, you have a 1/20 probability of winning $76 and a 19/20 probability of losing $9. What is your expected value?
Answer (1)
Calculate the expected value by multiplying each outcome by its probability.
The probability of winning $76 is 20 1 , and the probability of losing $9 is 20 19 .
Calculate the expected value: E = ( 20 1 ) × ( $76 ) + ( 20 19 ) × ( − $9 ) = 20 76 − 20 171 = − 20 95 .
The expected value is − $4.75 .
Explanation
Understand the problem and provided data We are given a game where there is a 20 1 probability of winning 76 an d a \frac{19}{20}$ probability of losing $9. We want to calculate the expected value of this game.
Define the expected value The expected value is calculated by multiplying each outcome by its probability and then summing the results. In this case, the expected value E is given by: E = ( Probability of winning ) × ( Amount won ) + ( Probability of losing ) × ( Amount lost )
Calculate the expected value Plugging in the given values, we have: E = ( 20 1 ) × ( $76 ) + ( 20 19 ) × ( − $9 ) E = 20 76 − 20 171 E = 20 76 − 171 E = 20 − 95 E = − 4 19 E = − $4.75
State the final answer The expected value of the game is − $4.75 . This means that, on average, you would expect to lose $4.75 each time you play the game.
Examples
Expected value calculations are used in various real-life scenarios, such as insurance, investment decisions, and gambling. For example, insurance companies use expected value to determine premiums, balancing the probability of a payout against the amount they expect to pay out. Similarly, investors use expected value to assess the potential profitability of different investments, considering both the potential gains and the risk of losses. In gambling, understanding the expected value helps players make informed decisions about whether to participate in a game, based on the potential payouts and the odds of winning.
