Fair Die Roll

What is the expected value when rolling a fair six-sided die?

List all outcomes and their probabilities: Outcomes: $1, 2, 3, 4, 5, 6$. Each has probability $\frac{1}{6}$ = 6 equally likely outcomes

Apply the expected value formula: $E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6}$ = Sum of weighted outcomes

Simplify the calculation: $E(X) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6) = \frac{21}{6}$ = $\frac{21}{6} = 3.5$

Answer: $E(X) = 3.5$. On average, you expect to roll 3.5 (even though you can never actually roll 3.5)!

Lottery Ticket

A lottery ticket costs 2 euros. You have a 1% chance of winning 100 euros and a 99% chance of winning nothing. What is the expected value of buying a ticket?

Identify outcomes and probabilities: Win: 100 euros with $P = 0.01$. Lose: 0 euros with $P = 0.99$ = Two possible outcomes

Calculate expected winnings: $E(\text{winnings}) = 100 \cdot 0.01 + 0 \cdot 0.99 = 1$ euro = Expected to win 1 euro

Subtract the cost of the ticket: $E(\text{profit}) = 1 - 2 = -1$ euro = Expected loss of 1 euro

Answer: $E(\text{profit}) = -1$ euro. On average, you lose 1 euro per ticket. This is why lotteries are profitable for the organizers!

Weighted Coin Game

A biased coin lands on heads 70% of the time. You win 5 euros for heads and lose 8 euros for tails. What is the expected value of one flip?

Assign values and probabilities: Heads: +5 euros, $P = 0.70$. Tails: $-8$ euros, $P = 0.30$ = One positive, one negative outcome

Apply the expected value formula: $E(X) = 5 \cdot 0.70 + (-8) \cdot 0.30$ = $E(X) = 3.5 + (-2.4)$

Calculate the final result: $E(X) = 3.5 - 2.4 = 1.1$ = $E(X) = 1.1$ euros

Answer: $E(X) = 1.1$ euros. On average, you win 1.10 euros per flip. This is a favorable game!

Insurance Expected Cost

An insurance company knows that 2% of customers will file a claim averaging 5000 euros. The annual premium is 150 euros. What is the company's expected profit per customer?

Calculate expected payout: $E(\text{payout}) = 5000 \cdot 0.02 + 0 \cdot 0.98 = 100$ euros = Expected to pay 100 euros per customer

Calculate expected profit: $E(\text{profit}) = \text{premium} - E(\text{payout})$ = $E(\text{profit}) = 150 - 100$

Find the answer: $E(\text{profit}) = 50$ euros per customer = 50 euros profit on average

Answer: The insurance company expects to make 50 euros profit per customer on average. This is how insurance companies stay in business!

Mistake: Forgetting that expected value can be a non-integer (like 3.5 on a die)

Why: Expected value is a theoretical average, not an actual outcome. You cannot roll 3.5, but over many rolls, your average will approach 3.5.

Correct: Understand that expected value represents the long-run average, not a specific outcome you will get.

Mistake: Using equal probabilities when outcomes are not equally likely

Why: A biased coin or weighted die has different probabilities for each outcome.

Correct: Always check if outcomes are equally likely. Use the actual probability for each outcome.

Mistake: Forgetting to subtract costs when calculating expected profit

Why: Winning 10 euros is not profit if you paid 15 euros to play!

Correct: $E(\text{profit}) = E(\text{winnings}) - \text{cost}$

Mistake: Adding probabilities instead of multiplying outcome by probability

Why: The formula is $x \cdot P(x)$, not $x + P(x)$.

Correct: Each outcome must be multiplied by its probability, then sum all the products.

Casino Games and House Edge

Casinos design games so that the expected value always favors the house. In roulette, the house edge means players lose about 2.7% (European) or 5.3% (American) of their bets on average.

In European roulette, betting 10 euros on red gives: $E = 10 \cdot \frac{18}{37} + (-10) \cdot \frac{19}{37} \approx -0.27$ euros. You lose 27 cents on average per 10-euro bet.

Investment Decisions

Investors use expected value to compare different investment options by weighing potential returns against their probabilities.

Stock A has a 60% chance of gaining 20% and a 40% chance of losing 10%. $E(\text{return}) = 0.20 \cdot 0.60 + (-0.10) \cdot 0.40 = 0.08 = 8\%$ expected return.

Game Show Strategy

Game show contestants use expected value to decide whether to keep a guaranteed prize or risk it for a chance at something bigger.

You have won 1000 euros. You can keep it or flip a coin: heads wins 2500 euros, tails wins nothing. $E(\text{flip}) = 2500 \cdot 0.5 + 0 \cdot 0.5 = 1250$ euros. Mathematically, you should flip!

Q: Why is the expected value of a die 3.5 if I can never roll 3.5?

A: Expected value is the average over many trials, not a single outcome. If you roll a die 1000 times and average all results, you'll get very close to 3.5. It's a theoretical center of the distribution.

Q: How is expected value different from the mode or median?

A: Expected value (mean) weights outcomes by probability. The mode is the most likely outcome, and the median is the middle value. For a fair die: mean = 3.5, mode = none (all equally likely), median = 3.5.

Q: Can expected value be negative?

A: Yes! If losses outweigh gains on average, the expected value is negative. Most gambling games have negative expected value for players, which is why casinos are profitable.

Q: Should I always choose the option with highest expected value?

A: Not necessarily. Expected value ignores risk tolerance. Would you bet your entire savings on a coin flip that doubles or loses it all? The expected value is positive, but most people wouldn't take that risk.