There are four $20 bills, six $10 bills, eight $5 bills and two $1 bills in your wallet. You randomly choose one bill


Is it possible for you to actually get the expected value when conducting this experiment?

To determine if it’s possible to actually get the expected value when randomly choosing one bill, we need to calculate the expected value first. The expected value (E) is the sum of all possible values multiplied by their respective probabilities.

### Step 1: Calculate the total number of bills
- 4 $\times$ $20 bills
- 6 $\times$ $10 bills
- 8 $\times$ $5 bills
- 2 $\times$ $1 bills

Total number of bills = $4 + 6 + 8 + 2 = 20$

### Step 2: Calculate the probability of each bill
- Probability of choosing a $20 bill: $\frac{4}{20} = \frac{1}{5}$
- Probability of choosing a $10 bill: $\frac{6}{20} = \frac{3}{10}$
- Probability of choosing a $5 bill: $\frac{8}{20} = \(\frac{2}{5}$
- Probability of choosing a $1 bill: $\frac{2}{20} = \(\frac{1}{10}$

### Step 3: Calculate the expected value
\[E = \left(\frac{4}{20} \times 20\right) + \left(\frac{6}{20} \times 10\right) + \left(\frac{8}{20} \times 5\right) + \left(\frac{2}{20} \times 1\right)\]

\[E = \left(\frac{80}{20}\right) + \left(\frac{60}{20}\right) + \left(\frac{40}{20}\right) + \left(\frac{2}{20}\right)\]

\[E = 4 + 3 + 2 + 0.1\]

\[E = 9.1\]

The expected value is $9.10.

### Step 4: Is it possible to get the expected value?
In this scenario, the expected value of $9.10 represents an average over many trials. However, you cannot physically draw a bill worth exactly $9.10 from your wallet, as you only have bills worth $20, $10, $5, and $1. The expected value is a theoretical mean that cannot be achieved in a single trial of the experiment.

Thus, **it is not possible** to actually get the expected value of $9.10 in one random draw. It only reflects what you would expect to average over many draws.

Would you like more details or have any questions?

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Here are 5 related questions:

1. How would the expected value change if you added more $1 bills?
2. What is the variance and standard deviation for this scenario?
3. How does the concept of expected value apply in real-world scenarios like lotteries?
4. What if you conducted this experiment multiple times—what would the distribution of your outcomes look like?
5. How does the expected value change if you remove one $20 bill?

**Tip:** When calculating expected values, remember that the expected value is not necessarily a value that can actually occur in a single trial; it’s an average over many trials.

Explore how to calculate the expected value when randomly selecting bills from a wallet containing $20 bills, $10 bills, $5 bills, and $1 bills. Understand the probability distributions and theoretical averages involved in this scenario. Suitable for high school students.

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