Question

Geoff is running a carnival game. He has 15 marbles in a bag: there are 3 green marbles, 7 red marbles and 5 yellow marbles. To play a round of the game,a player randomly takes out a marble from the bag, notes the color and replaces it, then pulls a second marble from the bag and notes the color. So in effect, the player pulls 2 marbles from the bag. (However, the first marble is put back in the bag and so potentially could be pulled twice.) Green marbles win 5 points, red marbles win 2 point and yellow marbles lose 2 points Let X be the random variable that describes the number of points won by a player playing a single round of Geoff's marble game. Find the probability distribution for X. Give values for X as whole numbers and probabilities as decimal values to 3 decimal places. Enter the values for X in ascending order (lowest to highest) from left to right in the table.

Answer 1

given below

Explanation:

given that he has 15 marbles in a bag: there are 3 green marbles, 7 red marbles and 5 yellow marbles.

To play a round of the game,a player randomly takes out a marble from the bag, notes the color and replaces it, then pulls a second marble from the bag and notes the color.

P(Green) =
`(3)/(15) =0.20`

P(Red) =
`(7)/(15) =0.467`

P(Yellow) =
`(5)/(15) =0.333`

since each time replaced the prob for I and II draw remains the same.

X=the random variable that describes the number of points won by a player playing a single round of Geoff's marble game.

If best he drawn 2 green there x=10, the worst is he draws two yellow and loses 4, if two red gain is 4 points

The different ways are 2 green, 2 red, 2 yellow, (1r,1g) (g,y) (y,r)

Events (y,y) (y,r) (y,g) (r,r) (r,g) (g,g) -- 6 number of ways

X = -4, 0 3 4 7 10

Pr 0.111 0.312 0.133 0.218 0.186 0.04

Whenever we have y,r i.e. change of colour we multiply by 2 because it can be y,r or r,y

The above is pdf of X