Question

Problem 4 (Bernoulli Trials) [25 marks] A multiple-choice exam con- sists of 20 questions. Each question offers 4 choices, one and only one of them being correct. The exam is passed if at least 16 questions are answered correctly. What is the probability that someone answering randomly passes the exam

Answer 1

This is a binomial distribution question with
n = 20
p = 0.25
q = 1 - p = 0.75
where

P(X = 1) =


P(X > 16), n = 20,p = 0.25 P(X > 16) = P(16<X<20) P(X > 16) = P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) P(X > 16) = (16) 0.25.180.7520–16+ (19)0.25"0.7520-17+(39) 0.25180.7520-15 + 20 19" 0.25190.7520–19 + 90 25200 7520-20 P(x > 16) = 4845.0(0.2516) (0.754)+1140.0(0.2517)(0.753)+190.0(0.2518)(0.752)+ 20.0(0.2519) (0.754) + 1(0.2520) (0.759) P(X > 16) = 4845.0(0.0) (0.3164)+1140.0(0.0) (0.4219)+190.0(0.0) (0.5625)+ 20.0(0.0) (0.75) + 100.0) (1.0) P(X > 16) = 3.569e-07 + 2.8e-08 +1.6e-09 + le-10 + 0.0 P(X > 16) = 3.865316102746874e-07

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