A game is played where two unbiased dice are rolled and the score in the game is thegreater of the two numbers shown. If the two numbers are the same, then the score in thegame is the number shown on one of the dice. A diagram showing the possible outcomes isgiven below.First die1 2 3 4 5 6Second die123456Let T be the random variable “the score in a game”.(a) Complete the table to show the probability distribution of T
Question
A game is played where two unbiased dice are rolled and the score in the game is thegreater of the two numbers shown. If the two numbers are the same, then the score in thegame is the number shown on one of the dice. A diagram showing the possible outcomes isgiven below.First die1 2 3 4 5 6Second die123456Let T be the random variable “the score in a game”.(a) Complete the table to show the probability distribution of T
Solution
The probability distribution of T can be calculated by considering all the possible outcomes when two dice are rolled.
There are a total of 36 possible outcomes when two dice are rolled (6 outcomes for the first die multiplied by 6 outcomes for the second die).
The possible scores in the game (T) are 1, 2, 3, 4, 5, and 6.
Let's calculate the probability for each score:
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T = 1: This can only happen if both dice show 1. So there is only 1 favorable outcome. The probability is 1/36.
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T = 2: This can happen if the dice show (1,2), (2,1), or (2,2). So there are 3 favorable outcomes. The probability is 3/36 = 1/12.
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T = 3: This can happen if the dice show (1,3), (2,3), (3,1), (3,2), or (3,3). So there are 5 favorable outcomes. The probability is 5/36.
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T = 4: This can happen if the dice show (1,4), (2,4), (3,4), (4,1), (4,2), (4,3), or (4,4). So there are 7 favorable outcomes. The probability is 7/36.
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T = 5: This can happen if the dice show (1,5), (2,5), (3,5), (4,5), (5,1), (5,2), (5,3), (5,4), or (5,5). So there are 9 favorable outcomes. The probability is 9/36 = 1/4.
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T = 6: This can happen if the dice show any combination with a 6. So there are 11 favorable outcomes. The probability is 11/36.
So the probability distribution of T is:
| T | P(T) |
|---|---|
| 1 | 1/36 |
| 2 | 1/12 |
| 3 | 5/36 |
| 4 | 7/36 |
| 5 | 1/4 |
| 6 | 11/36 |
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