need help with this question Suppose that a 4-face dice is rolled to determine h

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need help with this question

Suppose that a 4-face dice is rolled to determine how a certain amount of prize money is distributed among Student 1, Student 2 and Student 3. If the dice value X is 1, 2 or 3, Student X wins all prize money. If the dice value is 4, each student wins 1/3 of the prize money. Let A be the event that Student i wins nothing. (a) Find the probabilities of the three events A_1, A_2 and A_3. (b) Find the probabilities of the four events A_1A_2, A_1A_3, A_2A_3 and A_1A_2A_3. (c) Are A_1, A_2 and A_3 pairwise independent (each pair are independent)? (d) Are A_1, A_2 and A_3 independent?

Explanation / Answer

a) probabilty that one student wins nothing means his number or 4 does not appear hence probability =2/4 =1/2

P(A1)=P(A2)=P(A3)=1/2

b)for 2 student not to receive prize probabilty =1/4 cause that will happen when 3rd person will win all the prize

P(A1A2)=P(A2A3)=P(A3A1)=1/4

c)No as P(A1A2A3)=0 which is not equal to P(A1A2) *P(A2A3)

d) as P(A1A2A3)=0 is not equal to P(A1)*P(A2)*P(A3)

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