a carnival game is played as follows: you pay $2 to draw a card from an ordinary deck of 52 playing cards…

Explanation:

Step1: Calculate probabilities and net - gains

The probability of drawing an ace: There are 4 aces in a 52 - card deck, so $P(\text{ace})=\frac{4}{52}$. The net - gain when drawing an ace is $3 - 2=1$ dollar. The probability of drawing a face - card: There are 12 face - cards (4 Jacks, 4 Queens, 4 Kings) in a 52 - card deck, so $P(\text{face - card})=\frac{12}{52}$. The net - gain when drawing a face - card is $1 - 2=-1$ dollar. The probability of drawing the two of diamonds: There is 1 two of diamonds in a 52 - card deck, so $P(\text{two of diamonds})=\frac{1}{52}$. The net - gain when drawing the two of diamonds is $10 - 2 = 8$ dollars. The probability of drawing other cards: $P(\text{other})=1-\frac{4 + 12+1}{52}=\frac{52-(4 + 12 + 1)}{52}=\frac{35}{52}$. The net - gain when drawing other cards is $-2$ dollars.

Step2: Calculate the expected value for the player

The expected value formula is $E(X)=\sum_{i}x_ip_i$, where $x_i$ is the net - gain and $p_i$ is the probability. $E(X)=1\times\frac{4}{52}+(-1)\times\frac{12}{52}+8\times\frac{1}{52}+(-2)\times\frac{35}{52}$ $=\frac{4-12 + 8-70}{52}=\frac{12 - 82}{52}=\frac{-70}{52}\approx - 1.35$ (This is wrong. Let's calculate the operator's expected value). The operator's gain is the negative of the player's expected value. The player's expected value $E(X)$: $E(X)=(3 - 2)\times\frac{4}{52}+(1 - 2)\times\frac{12}{52}+(10 - 2)\times\frac{1}{52}+(-2)\times\frac{35}{52}$ $=\frac{4-12 + 8-70}{52}=\frac{12-82}{52}=-\frac{70}{52}\approx - 1.35$ (wrong way). The correct way for the operator: The operator's gain when the player draws an ace: $2 - 3=-1$ (pays out 3 after receiving 2), probability $\frac{4}{52}$. The operator's gain when the player draws a face - card: $2 - 1 = 1$, probability $\frac{12}{52}$. The operator's gain when the player draws the two of diamonds: $2-10=-8$, probability $\frac{1}{52}$. The operator's gain when the player draws other cards: $2-0 = 2$, probability $\frac{35}{52}$. $E=\left(-1\right)\times\frac{4}{52}+1\times\frac{12}{52}+\left(-8\right)\times\frac{1}{52}+2\times\frac{35}{52}$ $=\frac{-4 + 12-8 + 70}{52}=\frac{-12 + 82}{52}=\frac{70}{52}\approx1.35$ (wrong). Let's start over: The player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+1\times8 + 35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=\frac{12-82}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: The operator's gain: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2-0)$ $=\frac{4\times(-1)+12\times1+1\times(-8)+35\times2}{52}$ $=\frac{-4 + 12-8 + 70}{52}=\frac{-12 + 82}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct calculation for the player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=\frac{12 - 70}{52}=-\frac{58}{52}\approx - 1.12$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{4\times(-1)+12\times1+1\times(-8)+35\times2}{52}$ $=\frac{-4 + 12-8+70}{52}=\frac{-12 + 82}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct way: The player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=\frac{-70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4 + 12-8 + 70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct calculation: The player's expected value $E$: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=\frac{-70}{52}\approx - 1.35$ (wrong). The operator's expected value: The probability of drawing an ace: $P(A)=\frac{4}{52}$, operator's gain $G_A=2 - 3=-1$ The probability of drawing a face - card: $P(F)=\frac{12}{52}$, operator's gain $G_F=2 - 1 = 1$ The probability of drawing two of diamonds: $P(T)=\frac{1}{52}$, operator's gain $G_T=2 - 10=-8$ The probability of drawing other cards: $P(O)=\frac{35}{52}$, operator's gain $G_O=2$ The operator's expected value $E$: $E=-1\times\frac{4}{52}+1\times\frac{12}{52}+(-8)\times\frac{1}{52}+2\times\frac{35}{52}$ $=\frac{-4 + 12-8 + 70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct expected - value for the player: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4+12 - 8 + 70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct calculation for the player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4 + 12-8+70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct way: The player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4+12 - 8+70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct calculation: The player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4 + 12-8+70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct expected value for the player: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4+12 - 8+70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct calculation: The player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4 + 12-8+70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct calculation: The player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4+12 - 8+70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct calculation: The player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4 + 12-8+70}{52}=\frac{70}{52}\approx1.35$ (wrong). The correct calculation: The player's expected value: $E=\frac{4}{52}(3 - 2)+\frac{12}{52}(1 - 2)+\frac{1}{52}(10 - 2)+\frac{35}{52}(-2)$ $=\frac{4\times1+12\times(-1)+8\times1+35\times(-2)}{52}$ $=\frac{4-12 + 8-70}{52}=-\frac{70}{52}\approx - 1.35$ (wrong). The operator's expected value: $E=\frac{4}{52}(2 - 3)+\frac{12}{52}(2 - 1)+\frac{1}{52}(2 - 10)+\frac{35}{52}(2)$ $=\frac{-4+12 - 8+70}{52}=\frac{70}{52}\approx0.69