c) A die is rolled eight times and a 2, 3, 5 and 6 is considered a success. Find the probability of (i) At least 3 and at most 5 success. (ii) At least 2 successes. (iii) At most 1 but not any success. (iv) More than 6 but not all success
Answers
Given : A die is rolled eight times and a 2, 3, 5 and 6 is considered a success
To Find : probability of (i) At least 3 and at most 5 success. (ii) At least 2 successes. (iii) At most 1 but not any success. (iv) More than 6 but not all success
Solution:
A die is rolled eight times and a 2, 3, 5 and 6 is considered a success.
n ( E) = 4
n(S) = 6
Probability of success p = 4/6 = 2/3
probability of not success q = 1 - p = 1 - 2/3 = 1/3
P(x) = ⁿCₓpˣqⁿ⁻ˣ
n = 8 die is rolled eight times
At least 3 and at most 5 success.
= P(3) + P(4) + P(5)
= ⁸C₃(2/3)³(1/3)⁵ + ⁸C₄(2/3)⁴(1/3)⁴ + ⁸C₅(2/3)⁵(1/3)⁵
= 448/6561 + 1120/6561 + 1792/6561
= 3360/6561
= 1120/2187
At least 2 successes.
= P(2) + P(3) + P(4) + P(5) + P(6) + P(7) + P(8)
= 1 - P(0) - P(1)
= 1 - ( ⁸C₀(2/3)⁰(1/3)⁸ + ⁸C₁(2/3)¹(1/3)⁷ )
= 1 - ( 1 /6561 + 16/6561)
= 1 - (17/6561)
= 6544/6561
(iii) At most 1 but not any success. = 1 Succes
P(1) = ⁸C₁(2/3)¹(1/3)⁷ = 16/6561
iv) More than 6 but not all success = 6 & 7 Succees
P(6) + P(7)
= ⁸C₆(2/3)⁶(1/3)² + ⁸C₇(2/3)⁷(1/3)¹
= 1792/6561 + 1024/6561
= 2816/6561
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