Question

1. A certain casino claims that the commonly known phrase "The House Always Wins" does not apply to one of their games called "streak", In this game, players pay $2.00 for a chance to toss a coin until the outcome "tails" is observed for a maximum of four tosses (i.e., you can keep tossing until "tails" are observed). The casino pays the player $1.00 for each toss, so the longer your "streak" the better. Is the casino's argument right? (Hint: Argue for or against using the expected gain).

2 . In how many ways can the U.S. senate select a chairperson, a treasurer, and a secretary of a committee if all the senators are eligible for selection and no two senators from the same state can be selected together? (Hint: Remember from your Civics class that there are 100 senators in the U.S. senate – two per state!)

Answer 1

1)

here let X are number of flips in the game:

P(X=1) =P(1st flip tail) =1/2

P(X=2)=P(1st head and 2nd tails) =(1/2)² =1/4

P(X=3)=P(1st 2 head and 4rd tail) =(1/2)³ =1/8

P(X=4)=P(1st 3 head and 4th tail)+P(1st 4 heads) =(1/2)⁴+(1/2)⁴ =1/8

expected number of flips =E(X)=1*(1/2)+2*(1/4)+3*(1/8)+4*(1/8)=1.875

since expected value of the game to the player =expected win -fee to play =1.875*1-2 = -0.875

since expected value is negative, the claim made by casino is wrong,

2)

number of ways =N(select 3 state from 50 for each position and then there are 2 person to choose for each)

=(₅₀P₃)*2³ =19600*8 =156800