Suppose a teacher gives a multiple choice question with three answers [a, b and

Question

Suppose a teacher gives a multiple choice question with three answers [a, b and c]. An experiment consists of 2 student answering the question. Assuming they answer randomly, and each outcome is equally likely. a. Determine the sample space. b. Let E denote the event that "both students give the same answer", Define the event E. c. Let F denote the event that "at least one student answers b", Define the event F. d. Are the events E and F mutually exclusive? Why or why not? e. What is the probability that "at least one student answers b"? f. What is the probability that "at least one student answers b" or both student give the same answer? P(E or F) =P(E) + P(F) P(E or F) = P(E) + P(F) - P(E and F) P(E^C) = 1 - P(E) P(E and F) = P(E) middot P(F) P(E|F) = N(E and F)/N(F)

Explanation / Answer

Let the choice of answer for the first student be : a1,b1,c1

Let the choice of answer for the first student be : a2,b2,c2

a. Total sample space S : (a1,a2); (a1,b2); (a1,c2) ; (b1,a2); (b1,b2); (b1,c2) ; (c1,a2); (c1,b2); (c1,c2)

No. events = 9

b.Let E denote the event that "Both students give the same answers"

Event E : Both students can give the answer as "a", "b" or "c"

so the event space for E = (a1,a2);(b1,b2);(c1,c2)

c. Let F denote the event that "atleast one student answers "b"

Event F : Either student's 1 answer is "b" or students 2 answers is "b" or both the student's answer is b

Events Satsfying this condition are (b1, a2); (b1,b2) ; (b1,c2); (a1, b2); (c1,b2)

No. of events = 5

d. Are these events mutually exclusive ? No, these events are not mutually excluive.

because for event (b1,b2) satisfies both the criteria of E and F i.e both the answers are same and atleast one of the student's answers is b.

e. What is the probabilty of "atleast one student answer is b"

probabilty of "atleast one student answer is b" = P(F) = Number of events favoring F / Total number of events

From a . Total number of events = 9

From c. Number of events favoring F = 5

probabilty of "atleast one student answer is b" = P(F) = Number of events favoring F / Total number of events

P(F) = 5/9

f. The probability of "atleast one student answer is b" or " both the students give the same answer"

Event F : "atleast one student answer is b"

Event E " both the students give the same answer"

probability of "atleast one student answer is b" or " both the students give the same answer" = P(E or F) =

P(E or F) = P(E) + P(F) -P(F and E)

P(E) = Number of events favoring F / Total number of events = 3/9

probabilty of "atleast one student answer is b" = P(F) = 5/9

P(E and F) = Number of events favoring E and F / Total number of events =1/9

Number of events favoring E and F : 1 (event (b1,b2) satisfies both the criteria of E and F i.e both the answers are same and atleast one of the student's answers is b).

P(E or F) = P(E) + P(F) -P(F and E) = 3/9 + 5/9 -1/9 = 7/9

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