Question

In a carnival​ game, the player selects a ball one at a​ time, without​ replacement, from an urn containing twotwo purplepurple ballsballs and fourfour whitewhite ballsballs. the game proceeds until a purplepurple ball is drawn. the player pays ​$4.004.00 to play the game and receives ​$1.501.50 for each ball drawn. write down the probability distribution for the​ player's earnings, and find its expected value.

Answer 1

The urn contains 2 purple balls and 4 white balls. The player pay $4 for start the game and get $1.5 for every ball drawn until one purple ball is drawn. The maximal revenue would be $7.5 when 4 white balls and 1 purple balls are drawn.
If the purple ball is p and white ball is w, the possible sample space of drawings are {p, wp, wwp, wwwp, wwwwp}

1. Write down the probability distribution for the player earning

The player earning for each event depends on the number of balls drawn subtracted the ticket price.
p= 2/6
The player earnings would be: 1*$1.5 -$4= - $2.5
wp= (4*2)/(6*5) = 4/15
The player earnings would be: 2*1.5- $4= - $1
wwp= (4*3*2)/(6*5*4)= 1/5
The player earnings would be: 3*$1.5 -$4= $0.5
wwwp= (4*3*2*2)/(6*5*4*3*2)= 2/15 
The player earnings would be: 4*$1.5 -$4= $2
wwwwp= (4*3*2*2*1)/(6*5*4*3*2*1) = 1/15
The player earnings would be: 5*$1.5 -$4=  $3.5

2. Find its expected value

The expected value would be:
chance of event * earning 
You need to combine the 5 possible outcomes from the number 1 to get the total expected value.

Total expected value= (1/3 * - 2.5)+ (4/15*-1) + (1/5*0.5) + (2/15 *2) + ( 1/15 *3.5)= 
(-12.5  -4 + 1.5 + 4 + 3.5) /15= -$7.5
This game basically a rip off.