3. (a) A fair die is tossed eight times. Suppose A is the event of getting at le

Question

3. (a) A fair die is tossed eight times. Suppose A is the event of getting at lenst three 5's on the eight toeses of the die, while B is the event of getting precisely three 5's on the eight tosses of the die. i. Determine with reason if the events A and B are mutually exclusive. i. Determine the probabilities of the events A and B. Are the events A and B independent? (b) Suppose a fair coin is tossed 7 times. Let X be the discrete random variable representing the number of heads which can occur. i. Determiné the discrete probability distribution of X. ii. Determine the Expected value E(X) of X. ii. Determine the Variance, Var(x), of X. iv. Determine the Standard Deviation, a, of X.

Explanation / Answer

I am solving the question 3(a), since it contains two sub-parts, post multiple question to get the answer to the problem 3(b)

3(a)

(i)

For the events to be mutually exclusive, the events A and B must be disjoint

P(getting a 5 on die) = p = 1/6

P(not getting a 5) = q = 1 - p = 5/6

Event A = (there are three fives + there are four fives + ... + there are 8 gives)

Event B = (there are three fives)

Since one event is common in both A and B, which is occurence of three fives, hence the events are not mutually exculsive

b)

P(A) = 8C3(1/6)^3 (5/6)^5 + 8C4(1/6)^4*(5/6)^4 + ... + 8C8 * (1/6)^8 = 0.13484

P(B) = 8C3 * (1/5)^3 = 0.10419

For the sets to be independent, P(A int B) = P(A) P(B)

P(A int B) = P(B) = 0.10419, which is not equal to the product of P(A) * P(B)

Hence the sets are not linearly independent

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