Question

Eric has a bag with balls numbered through . he is playing a game of chance. this game is this: eric chooses one ball from the bag at random. he wins if the number is selected, if the number is selected, if the number is selected, if the number is selected, and if the number is selected. he loses if , , or is selected. (a) find the expected value of playing the game. (b) what can eric expect in the long run, after playing the game many times? (he replaces the ball in the bag each time.) eric can expect to gain money. eric can expect to lose money. eric can expect to break even (neither gain nor lose money).

Answer 1

To determine the worth of a game over the long run, the expected value is calculated by combining each outcome's probability with its respective gain or loss, giving the average financial impact of playing the game many times.

Step-by-step explanation:

To determine if the game of chance is financially worth playing over the long run, we calculate the expected value. This involves multiplying each possible outcome by its probability and summing up those products. The expected value reflects the average gain or loss per game if the game is played many times.

For the example with the 52-card deck and coin toss: the expected value is calculated using the probabilities of winning or losing given the card draw and coin result. With 12 face cards in a deck, and an equal chance of heads or tails, we combine these probabilities with the earnings or losses to find the expected value.

In the case of the game where matching all five numbers can lead to a large payout, we again calculate expected value by multiplying the profits ($100,000 or -$2) by their associated probabilities. The outcome with the large profit has a very low probability, which drastically affects the calculation.

Finally, for games with geometric distributions, we look at the probability of stopping the game after losing. If the probability of losing (a success in this context) is high, the expected number of games played until losing is typically lower.