games & expectation\n1. a box contains ten $1 bills, five $2 bills, three $5 bills, one $10 bill, and one…

games & expectation\n1. a box contains ten $1 bills, five $2 bills, three $5 bills, one $10 bill, and one $100 bill. a person is charged $20 to select one bill. find the expected value for this game. is this game fair?\n2. if a person rolls doubles when he tosses two dice, he wins $5. the cost to play the game is $1. is this game fair?

games & expectation\n1. a box contains ten $1 bills, five $2 bills, three $5 bills, one $10 bill, and one $100 bill. a person is charged $20 to select one bill. find the expected value for this game. is this game fair?\n2. if a person rolls doubles when he tosses two dice, he wins $5. the cost to play the game is $1. is this game fair?

Answer

Explanation:

Step1: Calculate the total number of bills

The total number of bills is (10 + 5+3 + 1+1=20)

Step2: Calculate the probability of each bill

  • Probability of ($1) bill: (P(1)=\frac{10}{20})
  • Probability of ($2) bill: (P(2)=\frac{5}{20})
  • Probability of ($5) bill: (P(5)=\frac{3}{20})
  • Probability of ($10) bill: (P(10)=\frac{1}{20})
  • Probability of ($100) bill: (P(100)=\frac{1}{20})

Step3: Calculate the expected value of the bill

The expected value (E(X)) of the bill is: [ \begin{align*} E(X)&=1\times\frac{10}{20}+2\times\frac{5}{20}+5\times\frac{3}{20}+10\times\frac{1}{20}+100\times\frac{1}{20}\ &=\frac{10 + 10+15 + 10+100}{20}\ &=\frac{145}{20}=7.25 \end{align*} ]

Step4: Calculate the expected value of the game

The cost to play the game is ($20). The expected value of the game (E) is (E = 7.25-20=- 12.75)

Answer:

The expected value of the game is (-$12.75). Since the expected value is not (0), the game is not fair.