Find Expected Value of a Game - Statistics
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Identify whether the game is favorable to the player if $E(\text{payoff})>0$.
Identify whether the game is favorable to the player if $E(\text{payoff})>0$.
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Favorable to the player. Positive expected payoff benefits the player.
Favorable to the player. Positive expected payoff benefits the player.
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State the formula for expected value $E(X)$ for outcomes $x_i$ with probabilities $p_i$.
State the formula for expected value $E(X)$ for outcomes $x_i$ with probabilities $p_i$.
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$E(X)=\sum x_i p_i$. Sum each outcome times its probability.
$E(X)=\sum x_i p_i$. Sum each outcome times its probability.
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What is the expected payoff if payoff $=X-c$ where $c$ is the ticket cost?
What is the expected payoff if payoff $=X-c$ where $c$ is the ticket cost?
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$E(\text{payoff})=E(X)-c$. Subtract cost from expected winnings.
$E(\text{payoff})=E(X)-c$. Subtract cost from expected winnings.
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Identify the fair-game condition in terms of expected payoff.
Identify the fair-game condition in terms of expected payoff.
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Fair game means $E(\text{payoff})=0$. Expected gain equals expected loss.
Fair game means $E(\text{payoff})=0$. Expected gain equals expected loss.
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What is the expected payoff if you win amount $w$ with probability $p$ and otherwise win $0$, with cost $c$?
What is the expected payoff if you win amount $w$ with probability $p$ and otherwise win $0$, with cost $c$?
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$pw-c$. Expected value of winning minus cost.
$pw-c$. Expected value of winning minus cost.
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Find the expected payoff: win $\$10$ with $p=\frac{1}{5}$, otherwise $0$; ticket costs $$1$.
Find the expected payoff: win $\$10$ with $p=\frac{1}{5}$, otherwise $0$; ticket costs $$1$.
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$1$. $E(\text{payoff})=\frac{1}{5}(10)-1=1$.
$1$. $E(\text{payoff})=\frac{1}{5}(10)-1=1$.
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Find the expected payoff: win $\$5$ with $p=\frac{1}{4}$, otherwise $0$; ticket costs $$2$.
Find the expected payoff: win $\$5$ with $p=\frac{1}{4}$, otherwise $0$; ticket costs $$2$.
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$-\frac{3}{4}$. $E(\text{payoff})=\frac{1}{4}(5)-2=-\frac{3}{4}$.
$-\frac{3}{4}$. $E(\text{payoff})=\frac{1}{4}(5)-2=-\frac{3}{4}$.
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Find the expected payoff: win $\$50$ with $p=0.02$, otherwise $0$; ticket costs $$1$.
Find the expected payoff: win $\$50$ with $p=0.02$, otherwise $0$; ticket costs $$1$.
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$0$. $E(\text{payoff})=0.02(50)-1=0$.
$0$. $E(\text{payoff})=0.02(50)-1=0$.
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Find $E(X)$: outcomes $\$0$ with $p=0.7$ and $$10$ with $p=0.3$.
Find $E(X)$: outcomes $\$0$ with $p=0.7$ and $$10$ with $p=0.3$.
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$3$. $E(X)=0.7(0)+0.3(10)=3$.
$3$. $E(X)=0.7(0)+0.3(10)=3$.
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Find $E(X)$: outcomes $\$-2$ with $p=\frac{3}{4}$ and $$6$ with $p=\frac{1}{4}$.
Find $E(X)$: outcomes $\$-2$ with $p=\frac{3}{4}$ and $$6$ with $p=\frac{1}{4}$.
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$0$. $E(X)=\frac{3}{4}(-2)+\frac{1}{4}(6)=0$.
$0$. $E(X)=\frac{3}{4}(-2)+\frac{1}{4}(6)=0$.
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Find the expected payoff: win $\$3$ with $p=\frac{1}{2}$, lose $$1$ with $p=\frac{1}{2}$.
Find the expected payoff: win $\$3$ with $p=\frac{1}{2}$, lose $$1$ with $p=\frac{1}{2}$.
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$1$. $E(\text{payoff})=\frac{1}{2}(3)+\frac{1}{2}(-1)=1$.
$1$. $E(\text{payoff})=\frac{1}{2}(3)+\frac{1}{2}(-1)=1$.
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A ticket costs $\$2$. Prizes: $$0$ with $0.8$, $\$5$ with $0.15$, $$20$ with $0.05$. Find expected payoff.
A ticket costs $\$2$. Prizes: $$0$ with $0.8$, $\$5$ with $0.15$, $$20$ with $0.05$. Find expected payoff.
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$-\frac{1}{4}$. $E(X)=0.8(0)+0.15(5)+0.05(20)=1.75$, minus cost $2$.
$-\frac{1}{4}$. $E(X)=0.8(0)+0.15(5)+0.05(20)=1.75$, minus cost $2$.
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Find the ticket cost $c$ for a fair game: win $\$12$ with $p=\frac{1}{6}$, otherwise $0$.
Find the ticket cost $c$ for a fair game: win $\$12$ with $p=\frac{1}{6}$, otherwise $0$.
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$2$. Set $\frac{1}{6}(12)-c=0$, solve for $c$.
$2$. Set $\frac{1}{6}(12)-c=0$, solve for $c$.
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Find the fair ticket cost $c$: win $\$5$ with $p=0.3$, otherwise $0$.
Find the fair ticket cost $c$: win $\$5$ with $p=0.3$, otherwise $0$.
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$1.5$. Set $0.3(5)-c=0$, solve for $c$.
$1.5$. Set $0.3(5)-c=0$, solve for $c$.
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Choose the correct method to compute expected payoff from a payoff table of $(x_i,p_i)$ values.
Choose the correct method to compute expected payoff from a payoff table of $(x_i,p_i)$ values.
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Compute $\sum x_i p_i$. Multiply outcomes by probabilities and sum.
Compute $\sum x_i p_i$. Multiply outcomes by probabilities and sum.
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Identify the expected payoff: win $\$4$ with $p=\frac{1}{3}$, win $$1$ with $p=\frac{2}{3}$ (no cost).
Identify the expected payoff: win $\$4$ with $p=\frac{1}{3}$, win $$1$ with $p=\frac{2}{3}$ (no cost).
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$2$. $E(\text{payoff})=\frac{1}{3}(4)+\frac{2}{3}(1)=2$.
$2$. $E(\text{payoff})=\frac{1}{3}(4)+\frac{2}{3}(1)=2$.
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Find expected payoff: prizes $\$0$ with $0.9$, $$10$ with $0.09$, $\$100$ with $0.01$; cost $$2$.
Find expected payoff: prizes $\$0$ with $0.9$, $$10$ with $0.09$, $\$100$ with $0.01$; cost $$2$.
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$-\frac{1}{10}$. $E(X)=0.9(0)+0.09(10)+0.01(100)=1.9$, minus $2$.
$-\frac{1}{10}$. $E(X)=0.9(0)+0.09(10)+0.01(100)=1.9$, minus $2$.
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What is the expected net gain if expected winnings are $E(W)$ and the ticket costs $c$?
What is the expected net gain if expected winnings are $E(W)$ and the ticket costs $c$?
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$E(\text{net})=E(W)-c$. Subtract the cost from expected winnings.
$E(\text{net})=E(W)-c$. Subtract the cost from expected winnings.
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State the formula for expected value $E(X)$ for outcomes $x_i$ with probabilities $p_i$.
State the formula for expected value $E(X)$ for outcomes $x_i$ with probabilities $p_i$.
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$E(X)=\sum x_i p_i$. Sum each outcome times its probability.
$E(X)=\sum x_i p_i$. Sum each outcome times its probability.
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What is the expected payoff if a game pays $a$ with probability $p$ and $b$ with probability $1-p$?
What is the expected payoff if a game pays $a$ with probability $p$ and $b$ with probability $1-p$?
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$E=ap+b(1-p)$. Multiply each payout by its probability and sum.
$E=ap+b(1-p)$. Multiply each payout by its probability and sum.
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Identify the fair ticket price $c$ in terms of expected winnings $E(W)$.
Identify the fair ticket price $c$ in terms of expected winnings $E(W)$.
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Fair price: $c=E(W)$. Fair when expected winnings equal cost.
Fair price: $c=E(W)$. Fair when expected winnings equal cost.
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What is the expected value of a constant random variable $X=k$?
What is the expected value of a constant random variable $X=k$?
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$E(X)=k$. Expected value of a constant is the constant itself.
$E(X)=k$. Expected value of a constant is the constant itself.
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State the linearity rule for expected value: $E(aX+b)$ in terms of $E(X)$.
State the linearity rule for expected value: $E(aX+b)$ in terms of $E(X)$.
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$E(aX+b)=aE(X)+b$. Scale by $a$ and shift by $b$.
$E(aX+b)=aE(X)+b$. Scale by $a$ and shift by $b$.
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A game pays $\$8$ with probability $\frac{1}{5}$ and costs $$1$ to play. Find expected net gain.
A game pays $\$8$ with probability $\frac{1}{5}$ and costs $$1$ to play. Find expected net gain.
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$\$0.60$. $E = 8 \cdot \frac{1}{5} - 1 = 1.6 - 1 = 0.60$
$\$0.60$. $E = 8 \cdot \frac{1}{5} - 1 = 1.6 - 1 = 0.60$
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Find $E(X)$ if $X$ is $\$20$ with probability $\frac{1}{50}$ and $$0$ otherwise.
Find $E(X)$ if $X$ is $\$20$ with probability $\frac{1}{50}$ and $$0$ otherwise.
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$\$0.40$. $E = 20 \cdot \frac{1}{50} + 0 \cdot \frac{49}{50} = 0.40$
$\$0.40$. $E = 20 \cdot \frac{1}{50} + 0 \cdot \frac{49}{50} = 0.40$
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What is the expected payoff for outcomes $\$3,\ $0,\ -$1$ with probabilities $0.2,\ 0.5,\ 0.3$?
What is the expected payoff for outcomes $\$3,\ $0,\ -$1$ with probabilities $0.2,\ 0.5,\ 0.3$?
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$\$0.3$. $E = 3(0.2) + 0(0.5) + (-1)(0.3) = 0.6 + 0 - 0.3$
$\$0.3$. $E = 3(0.2) + 0(0.5) + (-1)(0.3) = 0.6 + 0 - 0.3$
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Find the expected net gain if a ticket costs $\$2$ and pays $$10$ with probability $0.1$, else $\$0$.
Find the expected net gain if a ticket costs $\$2$ and pays $$10$ with probability $0.1$, else $\$0$.
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$-\$1$. $E = 10(0.1) + 0(0.9) - 2 = 1 - 2 = -1$
$-\$1$. $E = 10(0.1) + 0(0.9) - 2 = 1 - 2 = -1$
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What is the expected payoff if you win $\$5$ with probability $\frac{1}{4}$, otherwise $$0$?
What is the expected payoff if you win $\$5$ with probability $\frac{1}{4}$, otherwise $$0$?
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$\$1.25$. $E = 5 \cdot \frac{1}{4} + 0 \cdot \frac{3}{4} = 1.25$
$\$1.25$. $E = 5 \cdot \frac{1}{4} + 0 \cdot \frac{3}{4} = 1.25$
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What is the probability of winning if $1$ prize ticket is in a box of $N$ tickets?
What is the probability of winning if $1$ prize ticket is in a box of $N$ tickets?
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$\frac{1}{N}$. One winning ticket out of $N$ total tickets.
$\frac{1}{N}$. One winning ticket out of $N$ total tickets.
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A raffle has $200$ tickets at $\$2$ each and one prize of $$250$. Find expected net gain per ticket.
A raffle has $200$ tickets at $\$2$ each and one prize of $$250$. Find expected net gain per ticket.
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$-\$0.75$. $E = \frac{250}{200} - 2 = 1.25 - 2 = -0.75$
$-\$0.75$. $E = \frac{250}{200} - 2 = 1.25 - 2 = -0.75$
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