Question

In: Math

Geoff is running a carnival game. He has 15 marbles in a bag: there are 4...

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)

Solutions

Expert Solution

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

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