An organization runs a bingo night. You pay $5 for a card and get to play two games. For each game you win, you are paid $10. The table shows the probabilities of the three possible outcomes.
Win 0 Games; Probability= 0.81 
Win 1 Game; Probability= 0.18

Win 2 Games; Probability= 0.01

What is the standard deviation of the net result (“winnings”) for any participant in this bingo night, to the nearest cent?

To calculate the standard deviation of the net result (or "winnings") for a participant in this bingo night, we need to follow these steps:

### Step 1: Define the Random Variable
Let $ X $ be the random variable representing the net winnings of a participant. The possible outcomes for $ X $ are:
- Winning 0 games: Net result = $-5$ dollars (since you paid $5 and won nothing)
- Winning 1 game: Net result = $10 - 5 = 5$ dollars (since you paid $5 and won $10)
- Winning 2 games: Net result = $20 - 5 = 15$ dollars (since you paid $5 and won $20)

### Step 2: Calculate the Expected Value (Mean) $ E(X) $
First, we compute the expected value $ E(X) $ of the net result:

\[
E(X) = (-5 \times 0.81) + (5 \times 0.18) + (15 \times 0.01)
\]

\[
E(X) = (-4.05) + (0.9) + (0.15) = -3.00 \text{ dollars}
\]

### Step 3: Calculate the Variance
The variance $ \sigma^2 $ is calculated using:

\[
\sigma^2 = E(X^2) - (E(X))^2
\]

First, we calculate $ E(X^2) $:

\[
E(X^2) = (-5)^2 \times 0.81 + 5^2 \times 0.18 + 15^2 \times 0.01
\]

\[
E(X^2) = (25 \times 0.81) + (25 \times 0.18) + (225 \times 0.01)
\]

\[
E(X^2) = 20.25 + 4.5 + 2.25 = 27.00
\]

Next, we calculate the variance:

\[
\sigma^2 = 27.00 - (-3.00)^2 = 27.00 - 9.00 = 18.00
\]

### Step 4: Calculate the Standard Deviation
Finally, the standard deviation $ \sigma $ is the square root of the variance:

\[
\sigma = \sqrt{18.00} \approx 4.24 \text{ dollars}
\]

### Final Answer:
The standard deviation of the net result for any participant in this bingo night is approximately **$4.24**.

Do you want further details or have any other questions?

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Here are 5 related questions:
1. What is the expected net result for a participant in the bingo night?
2. How would the standard deviation change if the payout for winning both games increased to $25?
3. What is the probability that a participant will break even or make a profit?
4. How would the expected value change if the cost per card was $10 instead of $5?
5. How does increasing the number of participants affect the overall payout distribution?

**Tip:** Understanding how to calculate the standard deviation helps in assessing the risk or variability associated with different outcomes in probabilistic situations.

Learn how to calculate the standard deviation of net winnings for participants in a bingo night with a detailed step-by-step solution. Understand the concepts of probability, expected value, variance, and standard deviation applied to real-world scenarios. Suitable for advanced high school students and beyond.

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