Question

Please answer questions a b and c

I0 a) If a fair die is rolled 22 times, what is the probability that a 6 is obtained exactly 3 times? (b) If a fair die is rolled, what is the probability that the third time that a 6 is obtained is on the tenth roll? (c) If a fair coin is tossed 11 times, what is the prob ability that three or fewer heads are obtained?

Answer 1

Please note ⁿC_x = n! / [(n-x)!*x!]

(a) Binomial Probability = ⁿC_x * (p)^x * (q)^n-x, where n = number of trials and x is the number of successes.

Here n = 22, x = 3, p (Probability of getting a 6) = 1/6 and q = 1 - p = 1 - 1/6 = 5/6

P(3) = ²²C₃ * (1/6)³ * (5/6)^{22-3 = 19} = 0.2232

(b) Here since we are looking for the 3rd success (which is > 1 success). We use the negative Binomial Distribution

P[X = x] = ^x-1 C _k-1 * (p)^k * (q) ^x-k, where k is the success value, x is the number of trials.

Here x = 10 and k = 3. p = 1/6 and q = 5/6

Therefore P [X = 10] = ^10-1 C _3-1 x (1/6)³ x (5/6) ^{10-3 = 7} = 0.0465