Question

X
P(X=x)

Geoff is running a carnival game. He has 15 marbles in a bag: there are 4 green marbles, 7 red marbles and 4 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 1 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

As the possible marbles a player can draw from the bag in two turns are

{GG, GR, GY, RG, RR, RY, YG, YR, YY}

If the player draws green marbles in both turns, then he wins 5+5= 10 points.

Similarly we can find out for all the possible outcomes

Colour	GG	GR and RG	GY and YG	RR	RY and YR	YY
X	5+5=10	5+1=6	5-2=3	1+1=2	1-2=-1	-2-2=-4
P(X=x)

Now, probability of drawing a green marble in first turn is 4/15.

probability of drawing a green marble in second turn is 4/15.

Hence,

Similarly,

2 is multiplied because we will get 6 points in two ways one by having first green then red marble and second by having first red then second green marble.

In the similar ways we can find rest of the probabilities.