Ex 13.5, 7 - In an examination, 20 questions of true-false

Ex 13.5, 7 In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers 'true'; if it falls tails, he answers 'false'. Find the probability that he answers at least 12 questions correctly. Let X : be the number of questions he answers correctly Tossing a coin is a Bernoulli trial So, X has binomial distribution P(X = x) = nCx 𝒒﷮𝒏−𝒙﷯ 𝒑﷮𝒙﷯ n = number of questions = 10 p = Probability of getting answer correct Since it is a true false question P(he answers correctly) = p = 1﷮2﷯ q = 1 – p = 1 – 1﷮2﷯ = 1﷮2﷯ Hence, ⇒ P(X = x) = 20Cx 1﷮2﷯﷯﷮𝑥﷯ 1﷮2﷯﷯﷮20−𝑥﷯ P(X = x) = 20Cx 1﷮2﷯﷯﷮20 − 𝑥 + 𝑥﷯ P(X = x) = 20Cx 𝟏﷮𝟐﷯﷯﷮𝟐𝟎﷯ We need to find probability that he answers at least 12 questions correctly i.e. P(X ≥ 12) P(X ≥ 12) = P(12) + P(13) + P(14) + …… ….. +P(20) = 20C12 1﷮2﷯﷯﷮20﷯ + 20C13 1﷮2﷯﷯﷮20﷯+ 20C14 1﷮2﷯﷯﷮20﷯+ ……… + 20C20 1﷮2﷯﷯﷮20﷯ = 𝟏﷮𝟐﷯﷯﷮𝟐𝟎﷯(20C12 + 20C13 + 20C14 + …….. + 20C20)