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


c. What is the standard deviation of the net winnings? What does this value indicate?"

Verified step by step guidance

1. Step 1: Calculate the expected value (mean) of the net winnings. Use the formula for the expected value of a discrete random variable: \[\text{E}(X) = \sum (x_i \times p_i)\] where \(x_i\) is the profit for each outcome and \(p_i\) is the corresponding probability.
2. Step 2: Calculate the variance of the net winnings. Use the formula: \[\text{Var}(X) = \sum p_i (x_i - \mu)^2\] where \(\mu\) is the expected value calculated in Step 1, \(x_i\) are the profits, and \(p_i\) are the probabilities.
3. Step 3: Calculate the standard deviation by taking the square root of the variance: \[\sigma = \sqrt{\text{Var}(X)}\]
4. Step 4: Interpret the standard deviation value. It measures the typical amount by which the net winnings deviate from the expected value. A larger standard deviation indicates more variability or risk in the net winnings.
5. Step 5: Summarize your findings by stating the calculated standard deviation and explaining what it tells you about the variability of the net winnings in this video poker game.

Key Concepts

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of values. In the context of net winnings, it quantifies how much the actual winnings deviate from the expected value on average. A higher standard deviation indicates more variability and risk in the outcomes.

Expected Value

Expected value is the weighted average of all possible outcomes, calculated by multiplying each outcome's value by its probability and summing the results. It represents the long-term average profit or loss per bet in the video poker game.

Probability Distribution

A probability distribution lists all possible outcomes of a random experiment along with their probabilities. Understanding the distribution of profits and their probabilities is essential to calculate both the expected value and the standard deviation of net winnings.