Ex 13.5, 11

Transcript

Ex 13.5, 11 Find the probability of getting 5 exactly twice in 7 throws of a die. Let X : Number of times getting 5 Die thrown is a Bernoulli trial So, X has a binomial distribution P(X = x) = nCx 𝒒﷮𝒏−𝒙﷯ 𝒑﷮𝒙﷯ n = number of times die is thrown = 7 p = Probability of getting 5 = 1﷮6﷯ q = 1 – p = 1 – 1﷮6﷯ = 5﷮6﷯ Hence, ⇒ P(X = x) = 7Cx 𝟏﷮𝟔﷯﷯﷮𝒙﷯ 𝟓﷮𝟔﷯﷯﷮𝟕−𝒙﷯ We need to find Probability of getting 5 twice i.e. P(X = 2) Putting x = 2 in (1) P(X = 2) = 7C2 1﷮6﷯﷯﷮2﷯ 5﷮6﷯﷯﷮7−2﷯ = 7!﷮ 7 − 2﷯ ! 2 !﷯ 1﷮6﷯﷯﷮2﷯ 5﷮6﷯﷯﷮5﷯ = 7 !﷮5 ! 2 !﷯ 1﷮6 × 6﷯ 5﷮6﷯﷯﷮5﷯ = 7 × 6 × 5 !﷮5 ! 2 ! . 6 . 6﷯ 5﷮6﷯﷯﷮5﷯ = 7﷮2 × 1 × 6﷯ 5﷮6﷯﷯﷮2﷯= 7﷮12﷯ 5﷮6﷯﷯﷮5﷯ ∴ Required Probability = 𝟕﷮𝟏𝟐﷯ 𝟓﷮𝟔﷯﷯﷮𝟓﷯