Statistics Flashcards: Find Expected Value Of A Game

What this deck covers

This deck focuses on Find Expected Value Of A Game, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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