Question

O HE points T Premors Anss 8BUnderStat12 52.016 My Noles O AskYour Teacher Richard has just been given a 10-question multiple choice quiz in his history dass. Each question has five answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all ten questions, find the indicated probabilities. (Round your answers to three decimal places) (a) What is the probability that he will answer all questions correctly? (b) What is the probability that he will answer all questions incorrectly? (c) What is the probability that he will answer at least one of the questions correctiy? Compute this probability two ways. First, use the rule for mutually exdusive events and the probabilities shown, in the binomial probabilty distribution table Then use the fact that Pr2 1)-1-P 0) Compare the two results. Should they be equal? Are they equal? If not, how do you account fot the difference? They shouud be equal, but may differ slightly due to rounding emor They should be equal, but may not be due to table error They should be equal, but differ substantially They should not be equal, but are equal. dy what is the probability that Richard will answer at least halt the questions torrectiv?

Answer 1

Let X be the number of correctly answered questions

X ~ Binomial ( 10,0.2)

a) All questions are correct, X = 10

PlX-10)-㈡ = ( CO ) 0) (0.2)10(0.8)10-10-1.024 × 10-7

b) All Questions are incorrect , X = 0

PlX-0)-(,)(0.2)。(08)". 0.107374 0(0.8)10-0 0.107374

c)

P(X = 10)

The Calculation table for prbabilities are given below using the Excel Command

=BINOM.DIST(x,10,0.2,FALSE)

x	P(X=x)
1	0.268435
2	0.30199
3	0.201327
4	0.08808
5	0.026424
6	0.005505
7	0.000786
8	7.37E-05
9	4.1E-06
10	1.02E-07
Total	0.892626

P(X > 1) 0.8926

Also ,

P(X > 1) = 1-P(X = 0) = 1-0. 1074 = 0.8926

Comparision : They should be equal but may differ slightly due to rounding error

d)

P(X > 5) P(X-5) + P(X = 6) + + P(X-10)

PlX > 5) = 0.0264 + 0.0055 + + 1.024 × 10-,-0.03279