Question

ney invested. the weather forecast for the day of the concert is chance of rain. how much money can the community expect to make?
play a game where you draw one card from a standard deck of ng cards. if you pick a diamond, you win $10. if you pick a face 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?
bought one $1 raffle ticket to win a prize worth $500. if 1,000 s were sold and each ticket has an equal chance of winning, what is john’s expected gain?
cide to play a carnival game that offers these odds: you roll a ded die; if you roll an even number, you make $6. if you roll else, you lose $10. how much money do you expect to make playing this game?

Explanation:

Step1: Analyze the card game probabilities

A standard deck has 52 cards. Diamonds: 13 cards, face cards (not diamonds): 12 (3 face cards per non - diamond suit, 3 suits), other cards: \(52-(13 + 12)=27\) cards. Probabilities: \(P(\text{diamond})=\frac{13}{52}=\frac{1}{4}\), \(P(\text{face non - diamond})=\frac{12}{52}=\frac{3}{13}\), \(P(\text{other})=\frac{27}{52}\).

Step2: Calculate expected value

Expected value \(E\) is calculated as \(E=\sum x_iP(x_i)\). So \(E = 10\times\frac{13}{52}+6\times\frac{12}{52}+(- 5)\times\frac{27}{52}\).
First, calculate each term: \(10\times\frac{13}{52}=\frac{130}{52}\), \(6\times\frac{12}{52}=\frac{72}{52}\), \(-5\times\frac{27}{52}=-\frac{135}{52}\).
Then sum them: \(\frac{130 + 72-135}{52}=\frac{67}{52}\approx1.29\).

Answer:

The expected win is approximately \(\$1.29\) (or \(\frac{67}{52}\) dollars exactly).