Question

% chance of rain. how much money can the community expect to make?
ou play a game where you draw one card from a standard deck ofaying cards. if you pick a diamond, you win $10. if you pick a face
rd, which is not a diamond, you win $6. if you pick any other card,
you lose $5. how much money do you expect to win?
ohn bought one $1 raffle ticket to win a prize worth $500. if 1,000ickets were sold and each ticket has an equal chance of winning,
what is johns expected gain?
ou decide to play a carnival game that offers these odds: you roll a
six - sided die; if you roll an even number, you make $6. if you roll
ything else, you lose $10. how much money do you expect to make
playing this game?

Explanation:

Second Question (Card Game Expected Value)

Step1: Find Probabilities

A standard deck has 52 cards.

• Diamonds: 13 cards, so \( P(\text{diamond}) = \frac{13}{52} = \frac{1}{4} \).
• Face cards not diamonds: There are 3 face cards (J, Q, K) per suit, 3 suits (not diamonds), so \( 3 \times 3 = 9 \) cards. \( P(\text{face, not diamond}) = \frac{9}{52} \).
• Other cards: Total - diamonds - face non - diamonds = \( 52 - 13 - 9 = 30 \) cards. \( P(\text{other}) = \frac{30}{52} = \frac{15}{26} \).

Step2: Calculate Expected Value

Expected value \( E(X) = x_1P(x_1)+x_2P(x_2)+x_3P(x_3) \), where \( x_1 = 10 \), \( x_2 = 6 \), \( x_3=- 5 \).
\( E(X)=10\times\frac{1}{4}+6\times\frac{9}{52}+(-5)\times\frac{30}{52} \)
First, simplify each term:
\( 10\times\frac{1}{4}=\frac{10}{4}=\frac{5}{2} = 2.5 \)
\( 6\times\frac{9}{52}=\frac{54}{52}=\frac{27}{26}\approx1.038 \)
\( - 5\times\frac{30}{52}=-\frac{150}{52}=-\frac{75}{26}\approx - 2.885 \)
Now sum them: \( \frac{5}{2}+\frac{27}{26}-\frac{75}{26}=\frac{65 + 27-75}{26}=\frac{17}{26}\approx0.65 \)

Step1: Find Probabilities

Total tickets = 1000. Probability of winning \( P(\text{win})=\frac{1}{1000} \), probability of losing \( P(\text{lose}) = 1-\frac{1}{1000}=\frac{999}{1000} \).

Step2: Calculate Gains

• If win: Gain = \( 500 - 1=\$499 \) (since he spent \$1 on the ticket).
• If lose: Gain=\( - 1\) (he loses the \$1 he spent).

Step3: Calculate Expected Value

Expected gain \( E(X)=499\times\frac{1}{1000}+(-1)\times\frac{999}{1000} \)
\( E(X)=\frac{499 - 999}{1000}=\frac{- 500}{1000}=-0.5 \)

Step1: Find Probabilities

A six - sided die has 3 even numbers (2, 4, 6) and 3 odd numbers. So \( P(\text{even})=\frac{3}{6}=\frac{1}{2} \), \( P(\text{odd})=\frac{3}{6}=\frac{1}{2} \).

Step2: Calculate Expected Value

Expected value \( E(X)=6\times\frac{1}{2}+(- 10)\times\frac{1}{2} \)
\( E(X)=\frac{6 - 10}{2}=\frac{-4}{2}=-2 \)

Answer:

The expected amount to win is approximately \(\$0.65\) (or \(\frac{17}{26}\) dollars).

Third Question (Raffle Ticket Expected Gain)