Question

niceville community plans to invest $1,200 to host a concert. the community expects to sell tickets worth $1,600. but if it rains on the night of the concert, they wont sell any tickets, so they will lose all the money invested. the weather forecast for the day of the concert is 20% chance of rain. how much money can the community expect to make?
you play a game where you draw one card from a standard deck of playing cards. if you pick a diamond, you win $10. if you pick a face card, 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?
john bought one $1 raffle ticket to win a prize worth $500. if 1,000 tickets were sold and each ticket has an equal chance of winning, what is johns expected gain?
you 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 anything else, you lose $10. how much money do you expect to make playing this game?
$80 - $2.00 $0.65 - $0.50

Explanation:

First Problem (Concert Investment)

Step1: Calculate profit without rain

Profit without rain: \(1600 - 1200 = 400\) dollars. Probability of no rain: \(1 - 0.2 = 0.8\).

Step2: Calculate profit with rain

Profit with rain: \(-1200\) dollars (loss). Probability of rain: \(0.2\).

Step3: Calculate expected value

Expected value \(= (400 \times 0.8) + (-1200 \times 0.2) = 320 - 240 = 80\) dollars.

Step1: Determine probabilities

• Diamonds: 13 cards. Probability: \(\frac{13}{52} = \frac{1}{4}\).
• Non - diamond face cards: There are 9 non - diamond face cards (3 face cards per suit, 3 suits non - diamond). Probability: \(\frac{9}{52}\).
• Other cards: \(52-(13 + 9)=30\) cards. Probability: \(\frac{30}{52}=\frac{15}{26}\).

Step2: Calculate expected value

Expected value \(= 10\times\frac{1}{4}+6\times\frac{9}{52}+(-5)\times\frac{15}{26}\)
\(=\frac{10}{4}+\frac{54}{52}-\frac{75}{26}\)
\(=\frac{130 + 54 - 150}{52}=\frac{34}{52}\approx0.65\) dollars.

Step1: Determine probabilities

Probability of winning: \(\frac{1}{1000}\). Probability of losing: \(1-\frac{1}{1000}=\frac{999}{1000}\).

Step2: Calculate gains

Gain on winning: \(500 - 1=499\) dollars. Gain on losing: \(- 1\) dollar.

Step3: Calculate expected value

Expected value \(=499\times\frac{1}{1000}+(-1)\times\frac{999}{1000}=\frac{499 - 999}{1000}=\frac{- 500}{1000}=- 0.50\) dollars.

Answer:

\$80

Second Problem (Card Game)