Question 860623: Please help me solve this problem of probability.
We throw two fair dice. Let Random Variable that gives the outcome of i die, for i=1,2.
i) Define an appropriate space of probability that describes the above experiment.
ii) Find the probability and equation of distribution of X1.
iii) Find the probability of random variable Z=7-X1. What do you observe?
iv) Find the probability of random variable Z=X1+X2.
v) What is the the most probable value for the sum of the two dice?
vi) What is the average value of the sum of the two dice?
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! i) possible outcomes when rolling 2 dice, for each pair (a, b) a = first die, b = second die
1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
ii) X1 is a fair dice and possible outcomes are 1, 2, 3, 4, 5, 6. The probability of rolling any one of them is 1/6 and note that the sum of all six probalities must add up to 1, that is, the summation for i = 1 to 6 of Pi = 1
iii) Z=7-X1
P(7-1) = P(6) = 1/6, similarly P(i) where i = 1, 2...,6 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1
iv) this is a normal distribution for the sum X1 + X2, we list pairs of (sum, probability)
(2, 1/36), (3, 2/36), (4, 3/36), (5, 4/36), (6, 5/36), (7, 6/36), (8, 5/36), (9, 4/36), (10, 3/36), (11, 2/36), (12, 1/36)
v) from iv we can see that the probability of rolling a 7 is 6/36 = 1/6 which is the largest value
vi) the average or mean is 7, the graph of probabilities is a norm distribution with 7 in the middle (mean).
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