A student is taking a multiple-choice test in which each question has two choice

Question

A student is taking a multiple-choice test in which each question has two choices. She will draw one ball out of a bag with two balls in it and replaces that ball at the end of each question. The balls are marked such that she then knows what answer she will give. There are 10 multiple choice questions on the test. Given that she knows the correct answer to the first two questions, and she answers the other 8 questions randomly. A) What is the probability that she will get eight questions correct at the end of the test? B) What is the expected number of correct answers at the end of the test? Please show work.

Explanation / Answer

there are 10 multiple choice questions. out of which she knows the first two correctly, so there is nothing random in it.

she answers the other 8 questions randomly by drawing one ball out of a bag with two balls in it and replaces that ball at the end of each question. The balls are marked such that she then knows what answer she will give.

so the probability of correctly guessing an unknown answer is 0.5 and since she returns the ball after each answer the guessings are independent.

so if X denotes the number of correct answers given in the unknown 8 questions then X~Bin(8,0.5)

so the probability distribution of X is P[X=x]=8Cx0.58   x=0,1,2,3,...,8

a) the probability that she will get eight questions correct at the end of the test

=P[X=6] [as she would give two correct answers in the first two questions, so from the rest 8 unknowns 6 must be correct]

=8C60.58=0.109375 [answer]

b) since X ~Bin(8,0.2) so mean of X is E[X]=8*0.2=4

so expected number of correct answer in the 8 unknown asnwers is 4

ans she knows two answers correctly.

so expected number of correct answers at the end of the test is 4+2=6

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