Problem 1: A tetrahedrul die is a four-sided die with equally symmecric faces wi
ID: 3368429 • Letter: P
Question
Problem 1: A tetrahedrul die is a four-sided die with equally symmecric faces with numbers from 1 to 4 fair The other one is biased. The iased one lands on 4 with probability equal to and on the othet three mumbers ench with probability written on it. A person has s pair of tetrahedral die. One of them is A random experiment is performed where a pair such dioe are fipped i) Write down a sample space with the (ii) Use the probability measure to determine the probali ity of rolling six as ocrect probability measure to model this experiment. sum.Explanation / Answer
Solution
Let X = number on the top face of the fair die and Y = number on the top face of the biased die. Then, P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = ¼ …………………….(1) and
P(Y = 1) = P(Y = 2) = P(Y = 3) = 2/9, P(Y = 4) = 1/3 …………………….(2)
Back-up Theory
Since the number on the top face of the fair die does not depend on what number comes on the top face of the biased die, X and Y are independent ……………………..(4) and hence
P(X = x, Y = y) = P(X = x) x P(Y = y)…………………………………………..(5)
Part (I)
Vide (1), (2) and (5), the sample space alongwith the joint probability distribution of X and Y are given below:
x,p(x)
y,p(y)
Total
1, 2/9
2, 2/9
3, 2/9
4, 1/3
1, ¼
2/36
2/36
2/36
1/12
9/36 = ¼
2, ¼
2/36
2/36
2/36
1/12
9/36 = ¼
3, ¼
2/36
2/36
2/36
1/12
9/36 = ¼
4, ¼
2/36
2/36
2/36
1/12
9/36 = ¼
Total
8/36 = 2/9
8/36 = 2/9
8/36 = 2/9
8/36 = 2/9
1
DONE
Part (II)
Let S = sum of the numbers on the top face of the two dice. Then, S = X + Y.
Various possibilities, (x, y) of getting S = 6 and the respective probabilities are given below:
P(2, 4) = 1/12, P(3, 3) = 2/36, P(4, 2) = 2/36. Thus, P(S = 6) = (1/12) + (2/36) + (2/36)
= 7/36 ANSWER
x,p(x)
y,p(y)
Total
1, 2/9
2, 2/9
3, 2/9
4, 1/3
1, ¼
2/36
2/36
2/36
1/12
9/36 = ¼
2, ¼
2/36
2/36
2/36
1/12
9/36 = ¼
3, ¼
2/36
2/36
2/36
1/12
9/36 = ¼
4, ¼
2/36
2/36
2/36
1/12
9/36 = ¼
Total
8/36 = 2/9
8/36 = 2/9
8/36 = 2/9
8/36 = 2/9
1