"Video Poker The following table shows the net winnings from a \$1 bet in a video poker game.


b. If a player expects to play 90 games in one hour, how much can the player expect to win or lose during that hour?"

Verified step by step guidance

1. Step 1: Understand the problem context. The table provides the probabilities of different outcomes in a video poker game along with the corresponding profit (or loss) for each outcome from a \$1 bet.
2. Step 2: Calculate the expected profit for a single game. Use the formula for expected value: \(E(X) = \sum (P_i \times X_i)\), where \(P_i\) is the probability of outcome \(i\) and \(X_i\) is the profit for outcome \(i\). Multiply each profit by its probability and sum all these products.
3. Step 3: Interpret the expected value. The expected value represents the average profit or loss per game if the game is played many times.
4. Step 4: Calculate the expected profit for 90 games. Since the player expects to play 90 games in one hour, multiply the expected profit per game by 90 to find the total expected profit or loss for the hour.
5. Step 5: Summarize the result. The final value from step 4 will indicate how much the player can expect to win or lose during one hour of playing 90 games.

Key Concepts

Expected Value

Expected value is the weighted average of all possible outcomes, calculated by multiplying each outcome's value by its probability and summing these products. It represents the long-term average gain or loss per game, helping players understand the average result of repeated plays.

Probability Distribution

A probability distribution lists all possible outcomes of a random experiment along with their probabilities. Understanding this distribution is essential to calculate expected values and to assess the likelihood of different results in games like video poker.

Scaling Expected Value Over Multiple Trials

When a game is played multiple times, the total expected winnings or losses are found by multiplying the expected value per game by the number of games played. This helps estimate overall profit or loss over a session, such as 90 video poker games in one hour.