Find Expected Value of a Game - Statistics and Probability

1. Find expected payoff: prizes $0 with $0.9$, $10 with $0.09$, $100 with $0.01$; cost $2.

   $-\frac{1}{10}$. $E(X)=0.9(0)+0.09(10)+0.01(100)=1.9$, minus $2$.

2. A game pays $4 with probability $0.3$ and $1 with probability $0.7$. Find expected payoff.

   $1.90. $E = 4(0.3) + 1(0.7) = 1.2 + 0.7 = 1.9$

3. A fast-food promo gives $0 with probability $0.8$ and $5 with probability $0.2$. Find $E(W)$.

   $1. $E = 0(0.8) + 5(0.2) = 0 + 1 = 1$

4. State the formula for expected value $E(X)$ for outcomes $x_i$ with probabilities $p_i$.

   $E(X)=\sum x_i p_i$. Sum each outcome times its probability.

5. What is the expected payoff for outcomes $\$3,\ \$0,\ -\$1$ with probabilities $0.2,\ 0.5,\ 0.3$?

   $0.3. $E = 3(0.2) + 0(0.5) + (-1)(0.3) = 0.6 + 0 - 0.3$

6. What is the expected payoff if you win $5 with probability $\frac{1}{4}$, otherwise $0?

   $1.25. $E = 5 \cdot \frac{1}{4} + 0 \cdot \frac{3}{4} = 1.25$

7. Find the expected payoff: win $10 with $p=\frac{1}{5}$, otherwise $0$; ticket costs $1.

   $1$. $E(\text{payoff})=\frac{1}{5}(10)-1=1$.

8. Find the expected payoff: win $5 with $p=\frac{1}{4}$, otherwise $0$; ticket costs $2.

   $-\frac{3}{4}$. $E(\text{payoff})=\frac{1}{4}(5)-2=-\frac{3}{4}$.

9. Identify the expected payoff: win $4 with $p=\frac{1}{3}$, win $1 with $p=\frac{2}{3}$ (no cost).

   $2$. $E(\text{payoff})=\frac{1}{3}(4)+\frac{2}{3}(1)=2$.

10. Identify the fair ticket price $c$ in terms of expected winnings $E(W)$.

    Fair price: $c=E(W)$. Fair when expected winnings equal cost.

11. Find the expected payoff: win $50 with $p=0.02$, otherwise $0$; ticket costs $1.

    $0$. $E(\text{payoff})=0.02(50)-1=0$.

12. Find the expected net gain if a ticket costs $2 and pays $10 with probability $0.1$, else $0.

    $-\$1$. $E = 10(0.1) + 0(0.9) - 2 = 1 - 2 = -1$

13. What is the expected net gain if expected winnings are $E(W)$ and the ticket costs $c$?

    $E(\text{net})=E(W)-c$. Subtract the cost from expected winnings.

14. Choose the correct method to compute expected payoff from a payoff table of $(x_i,p_i)$ values.

    Compute $\sum x_i p_i$. Multiply outcomes by probabilities and sum.

15. State the linearity rule for expected value: $E(aX+b)$ in terms of $E(X)$.

    $E(aX+b)=aE(X)+b$. Scale by $a$ and shift by $b$.

16. What is the probability of winning if $1$ prize ticket is in a box of $N$ tickets?

    $\frac{1}{N}$. One winning ticket out of $N$ total tickets.

17. Find the expected payoff: win $3 with $p=\frac{1}{2}$, lose $1 with $p=\frac{1}{2}$.

    $1$. $E(\text{payoff})=\frac{1}{2}(3)+\frac{1}{2}(-1)=1$.

18. Find the fair ticket cost $c$: win $5 with $p=0.3$, otherwise $0$.

    $1.5$. Set $0.3(5)-c=0$, solve for $c$.

19. State the formula for expected value $E(X)$ for outcomes $x_i$ with probabilities $p_i$.

    $E(X)=\sum x_i p_i$. Sum each outcome times its probability.

20. What is the expected payoff if you win amount $w$ with probability $p$ and otherwise win $0$, with cost $c$?

    $pw-c$. Expected value of winning minus cost.

21. Choose the expected net gain if cost is $1 and payout is $3 with probability $\frac{1}{2}$, else $0.

    $0.50. $E = 3 \cdot \frac{1}{2} - 1 = 1.5 - 1 = 0.50$

22. A game pays $6 with probability $\frac{1}{3}$, $0 with probability $\frac{2}{3}$. Find fair price.

    $2. Fair price equals expected value: $6 \cdot \frac{1}{3} = 2$

23. Find the ticket cost $c$ for a fair game: win $12 with $p=\frac{1}{6}$, otherwise $0$.

    $2$. Set $\frac{1}{6}(12)-c=0$, solve for $c$.

24. Find $E(X)$: outcomes $-2 with $p=\frac{3}{4}$ and $6 with $p=\frac{1}{4}$.

    $0$. $E(X)=\frac{3}{4}(-2)+\frac{1}{4}(6)=0$.

25. What is the expected payoff if payoff $=X-c$ where $c$ is the ticket cost?

    $E(\text{payoff})=E(X)-c$. Subtract cost from expected winnings.

26. What is the expected value of a constant random variable $X=k$?

    $E(X)=k$. Expected value of a constant is the constant itself.

27. A raffle has $200$ tickets at $2 each and one prize of $250. Find expected net gain per ticket.

    $-\$0.75$. $E = \frac{250}{200} - 2 = 1.25 - 2 = -0.75$

28. Find the missing probability $p$ if outcomes are $5 with probability $p$ and $0 with probability $0.6$.

    $p=0.4$. Probabilities sum to $1$: $p + 0.6 = 1$

29. Identify whether the game is favorable: cost $2, payout $9 with probability $0.2$, else $0.

    Unfavorable since $E(\text{net})=-\$0.20$. $E = 9(0.2) - 2 = 1.8 - 2 = -0.20 < 0$

30. What is the expected payoff if a game pays $a$ with probability $p$ and $b$ with probability $1-p$?

    $E=ap+b(1-p)$. Multiply each payout by its probability and sum.

31. Identify whether the game is favorable to the player if $E(\text{payoff})>0$.

    Favorable to the player. Positive expected payoff benefits the player.

32. A ticket costs $2. Prizes: $0 with $0.8$, $5 with $0.15$, $20 with $0.05$. Find expected payoff.

    $-\frac{1}{4}$. $E(X)=0.8(0)+0.15(5)+0.05(20)=1.75$, minus cost $2$.

33. A game pays $8 with probability $\frac{1}{5}$ and costs $1 to play. Find expected net gain.

    $0.60. $E = 8 \cdot \frac{1}{5} - 1 = 1.6 - 1 = 0.60$

34. Find $E(X)$ if $X$ is $20 with probability $\frac{1}{50}$ and $0 otherwise.

    $0.40. $E = 20 \cdot \frac{1}{50} + 0 \cdot \frac{49}{50} = 0.40$

35. A prize box has $3$ cards worth $2 and $7$ cards worth $0. Find expected payoff per draw.

    $0.60. $E = \frac{3 \cdot 2 + 7 \cdot 0}{10} = \frac{6}{10} = 0.60$

36. Find $E(X)$: outcomes $0 with $p=0.7$ and $10 with $p=0.3$.

    $3$. $E(X)=0.7(0)+0.3(10)=3$.

37. Identify the fair-game condition in terms of expected payoff.

    Fair game means $E(\text{payoff})=0$. Expected gain equals expected loss.

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