Question

At a casino, people line up to pay $20 each to be a contestant in the follow- ing game: The contestant flips a fair coin repeatedly. If she flips heads 20 times in 20 mil a row, she walks away with lion dollars; otherwise she walks away with R 0 dollars

(a) Find the PMF of R, the reward earned by the contestant.
(b) The casino counts "losing contestants who fail to win the 20 million dollar prize. Let L equal the number of los- ing contestants before the first winning contestant. What is the PMF of L?
(c) Why does the casino offer this game?

Answer 1

a)

P ( R = 20 million ) = 0.00000095367

P ( R = 0 million ) i.e. not winning = 0.99999904633

b) P ( L ) = ( 1 - p )^l * p

C)The casino offer the game because the expected revenue exceeds the winning

Explanation:

P( winning ) = P ( getting 20heads in a row ) = 0.5^20 = 0.00000095367

a) PMF of R

P ( R = 20 million ) = 0.00000095367

P ( R = 0 million ) i.e. not winning = 0.99999904633

b) what is the PMF of L

L = number of loosing clients before winning

P( L = 0 ) = P i.e. if first contestant wins, if the second contestant win, L=1 repeat same process continuously

P ( L = 2 ) = ( 1 - p)^2 p

Hence the PMF of L

P ( L ) = ( 1 - p )^l * p where p = 0.5^20

C) The casino offer the game because the expected revenue exceeds the winning

we can prove this using the relation below

E( L ) = ( 1 - P ) / 1 - ( 1 - P )

= 1 / P - 1

where p = 0.00000095367 ( probability of winning )

Hence E ( L ) = 1048575

hence expected revenue before someone wins

= (1048575 + 1 ) * 20 = $20,971,520