Misc 9

Transcript

Misc 9 An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes. Let X: Number of successes Since we are talking about success and failure It is a Bernoulli trial So, X has a binomial distribution n = number of trials = 6 p = Probability of success q = Probability of failure = 1 – p Given experiment succeeds twice as often it fails So, p = 2q p = 2(1 – p) p = 2(1 – p) p = 2 – 2p 3p = 2 p = 𝟐﷮𝟑﷯, Hence, q = 1 – p = 1 – 2﷮3﷯ = 1﷮3﷯ Hence, P(X = x) = 6Cx 𝟐﷮𝟑﷯﷯﷮𝒙﷯ 𝟏﷮𝟑﷯﷯﷮𝟔−𝒙﷯ We need to probability that there will be at least 4 successes i.e. P(X ≥ 4) P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) = 6C4 2﷮3﷯﷯﷮4﷯ 1﷮3﷯﷯﷮6−4﷯+6C5 2﷮3﷯﷯﷮5﷯ 1﷮3﷯﷯﷮6−5﷯+6C6 2﷮3﷯﷯﷮6﷯ 1﷮3﷯﷯﷮6−6﷯ = 6C4 2﷮3﷯﷯﷮4﷯ 1﷮3﷯﷯﷮2﷯+6C5 2﷮3﷯﷯﷮5﷯ 1﷮3﷯﷯﷮1﷯+6C6 2﷮3﷯﷯﷮6﷯ 1﷮3﷯﷯﷮0﷯ = 15 × 2﷮3﷯﷯﷮4﷯ 1﷮3﷯﷯﷮2﷯+6 2﷮3﷯﷯﷮5﷯ 1﷮3﷯﷯﷮1﷯+1 2﷮3﷯﷯﷮6﷯ × 1 = 2﷮3﷯﷯﷮4﷯ 15 × 1﷮3﷯﷯﷮2﷯+6 2﷮3﷯﷯ 1﷮3﷯﷯+ 2﷮3﷯﷯﷮2﷯﷯ = 2﷮3﷯﷯﷮4﷯ 15﷮9﷯+ 12﷮9﷯+ 4﷮9﷯﷯ = 𝟑𝟏﷮𝟗﷯ 𝟐﷮𝟑﷯﷯﷮𝟒﷯