"Video Poker The following table shows the net winnings from a \$1 bet in a video poker game.
a. Calculate and explain the expected net winnings from the player's perspective. Round your answer to three decimal places (nearest tenth of a penny).
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Verified step by step guidance

1. Identify the random variable and its possible values: Here, the random variable is the net winnings (profit) from a \$1 bet in video poker, with each outcome having a specific profit value and associated probability.
2. Recall the formula for the expected value (mean) of a discrete random variable: \(\displaystyle E(X) = \sum (x_i \times p_i)\), where \(x_i\) is the profit for outcome \(i\) and \(p_i\) is the probability of that outcome.
3. Multiply each profit value by its corresponding probability to find the weighted value for each outcome. For example, for Royal Flush, calculate \$799 \times 0.000023\(, for Straight Flush, calculate \)49 \times 0.000142$, and so on for all outcomes.
4. Sum all the weighted values obtained in the previous step to get the expected net winnings: \(E(X) = (799 \times 0.000023) + (49 \times 0.000142) + \ldots + (-1 \times 0.58076)\).
5. Interpret the result: The expected value represents the average net winnings per \$1 bet over a large number of plays. A positive expected value means the player expects to win on average, while a negative value means an expected loss.

Key Concepts

Expected Value

Expected value is the weighted average of all possible outcomes, where each outcome is multiplied by its probability. It represents the long-term average result of a random event, such as net winnings in a game. Calculating expected value helps determine whether a game is favorable or not.

Probability Distribution

A probability distribution lists all possible outcomes of a random experiment along with their probabilities. In this context, it shows the likelihood of each poker hand occurring. Understanding the distribution is essential to correctly compute the expected value.

Net Winnings (Profit) in Gambling

Net winnings refer to the profit or loss from a bet after accounting for the initial stake. Positive values indicate gains, while negative values indicate losses. Analyzing net winnings alongside probabilities allows players to assess the risk and reward of the game.