Business Mathematics I - Expected Value Worksheet 2 | MATH 115A ... - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

Problem 1

Goal: Find the average win or loss per game.

Step 1: Determine all possible outcomes.
When tossing three coins, there are $2 \times 2 \times 2 = 8$ total possible outcomes (like HHH, HHT, etc.).

Step 2: Calculate probabilities and values for each case.
* Exactly two heads: The outcomes are HHT, HTH, THH. There are 3 outcomes.
* Probability: $\frac{3}{8}$
* Value: Win $\$5$ ($+5$)
* Exactly one head: The outcomes are HTT, THT, TTH. There are 3 outcomes.
* Probability: $\frac{3}{8}$
* Value: Win $\$3$ ($+3$)
* Otherwise (0 heads or 3 heads): The outcomes are TTT and HHH. There are 2 outcomes.
* Probability: $\frac{2}{8}$
* Value: Lose $\$2$ ($-2$)

Step 3: Calculate the Expected Value ($E$).
Multiply each value by its probability and add them up:
$$E = (5 \cdot \frac{3}{8}) + (3 \cdot \frac{3}{8}) + (-2 \cdot \frac{2}{8})$$
$$E = \frac{15}{8} + \frac{9}{8} - \frac{4}{8}$$
$$E = \frac{20}{8} = 2.5$$

Interpretation: Since the result is positive, the student wins money on average.

***

Problem 2

Goal: Find the fee required to cover costs on average.

Step 1: Identify the variables.
* Cost of investigation: $\$9000$ (This is a guaranteed loss, so $-9000$).
* Probability of recovering property (Success): $\frac{1}{9}$.
* Probability of failure: $\frac{8}{9}$.
* Let $F$ be the fee charged. The detective only gets paid if he succeeds.

Step 2: Set up the Expected Value equation.
To "break even" or cover costs, the Expected Value must be at least 0.
$$E = (\text{Winning Amount} \cdot P(\text{Win})) + (\text{Losing Amount} \cdot P(\text{Loss}))$$

If he wins, he gets the Fee ($F$) but paid the cost ($9000$), so net is $F - 9000$.
If he loses, he gets $\$0$ but paid the cost ($9000$), so net is $-9000$.

$$0 = (F - 9000) \cdot \frac{1}{9} + (-9000) \cdot \frac{8}{9}$$

Step 3: Solve for $F$.
Multiply everything by 9 to clear the fractions:
$$0 = (F - 9000) \cdot 1 + (-9000) \cdot 8$$
$$0 = F - 9000 - 72000$$
$$0 = F - 81000$$
$$F = 81000$$

*(Alternative simpler logic: To cover a $\$9000$ cost with only a $\frac{1}{9}$ chance of getting paid, you need to charge 9 times the cost: $9000 \times 9 = 81000$.)*

***

Problem 3

Goal: Calculate expected value and decide if it's worthwhile.

Step 1: List probabilities and net winnings.
Note: You must subtract the $\$1000$ entry fee from every prize to find the actual "net" profit.
* 1st Place: Prob $\frac{1}{20}$. Prize $\$4500$. Net: $4500 - 1000 = \$3500$.
* 2nd Place: Prob $\frac{1}{10}$. Prize $\$3500$. Net: $3500 - 1000 = \$2500$.
* 3rd Place: Prob $\frac{1}{4}$. Prize $\$1500$. Net: $1500 - 1000 = \$500$.
* No Place: We need to find this probability.
$$1 - (\frac{1}{20} + \frac{1}{10} + \frac{1}{4}) = 1 - (\frac{1}{20} + \frac{2}{20} + \frac{5}{20}) = 1 - \frac{8}{20} = \frac{12}{20}$$
Net: $-\$1000$ (You lose your entry fee).

Step 2: Calculate Expected Value ($E$).
$$E = (3500 \cdot \frac{1}{20}) + (2500 \cdot \frac{2}{20}) + (500 \cdot \frac{5}{20}) + (-1000 \cdot \frac{12}{20})$$
*(Note: I converted $\frac{1}{10}$ to $\frac{2}{20}$ and $\frac{1}{4}$ to $\frac{5}{20}$ for easier math).*

$$E = \frac{3500}{20} + \frac{5000}{20} + \frac{2500}{20} - \frac{12000}{20}$$
$$E = \frac{3500 + 5000 + 2500 - 12000}{20}$$
$$E = \frac{11000 - 12000}{20}$$
$$E = \frac{-1000}{20} = -50$$

Interpretation: The expected value is $-\$50$. This means on average, you lose $\$50$ every time you race. Therefore, it is not worthwhile.

***

Problem 4

Goal: Find the ticket price to break even.

Step 1: Calculate the total expected payout per ticket.
The club needs to collect enough money in tickets to pay out the prizes on average.
* Prize 1: $\$100$ with prob $0.002$.
Value: $100 \cdot 0.002 = \$0.20$
* Prize 2: $\$80$ with prob $0.01$.
Value: $80 \cdot 0.01 = \$0.80$

Step 2: Add the expected payouts together.
Total Expected Payout = $\$0.20 + \$0.80 = \$1.00$.

Step 3: Determine Ticket Price.
To break even, the ticket price must equal the expected payout.

Final Answer:
1. On average, he should win $2.50 per play.
2. His fee should be $81,000.
3. The expected value is -$50. It is not worthwhile because the expected value is negative, meaning you will lose money on average.
4. The club should charge $1.00 for tickets.