Question

An exam consists of 10 true-or-false questions. If a student guesses at every answer, what is the probability that he or she will answer exactly 6 questions correctly? (Round your answer to three decimal places.)

Answer 1

X ~ Bin(n,p)

Where n = 10 , p = 0.5

np = 10 * 0.5 = 5 >= 5

n(1-p) = 10 * 0.5 = 5 >= 5

Since np >= 5 and n(1-p) >= 5 , normal approximation is appropriate.  

Mean = np = 10 * 0.5 = 5

Standard deviation = sqrt [ np(1-p) ]

= sqrt [ 10 * 0.5 * 0.5 ]

= 1.5811

Using normal approximation,

P(X < x) = P(Z < x - Mean / SD)

With continuity correction,

P(X = 6) = P(5.5 < X < 6.5)

= P(X < 6.5) - P(X < 5.5)

= P(Z < (6.5 - 5) / 1.5811) - P(Z < (5.5 - 5) / 1.5811)

= P(Z < 0.95) - P(Z < 0.32)

= 0.8289 - 0.6255

= 0.203