Question

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 2 marbles (without replacement) from the bag. 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.

X
P(X=x)

Answer 1

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

The player randomly takes out 2 marbles (without replacement), all the combination for 2 marbles are,

Combination	Marble 1	Marble 2
1	Green	Green
2	Green	Red
3	Green	Yellow
4	Red	Green
5	Red	Red
6	Red	Yellow
7	Yellow	Green
8	Yellow	Red
9	Yellow	Yellow

Since some combination are repeated, the distinct combinations are,

Combination	Marble 1	Marble 2
1	Green	Green
2	Green	Red
	Red	Green
3	Green	Yellow
	Yellow	Green
4	Red	Red
5	Red	Yellow
	Yellow	Red
6	Yellow	Yellow

The probability of selecting two marbles is obtained as follow,

Using the conditional probability law,

Green-Green

Green-Red or Red-Green

Green-Yellow or Yellow-Green

Red-Red

Red-Yellow or Yellow-Red

Yellow-Yellow,

The payoff for each combination is(in ascending order,

Combination	Marble 1-Marble 2	Payoff
1	YY	-4
2	RY/YR	-1
3	RR	2
4	GY/YG	3
5	GR/RG	6
6	GG	10

The distribution of random variable X is,

X	P(X=x)
-4	0.057
-1	0.267
2	0.200
3	0.152
6	0.267
10	0.057