Suppose that a die is loaded so that {1, 3, 4, 6} each come up equally often as

Question

Suppose that a die is loaded so that {1, 3, 4, 6} each come up equally often as each other, 2 comes up three times as often {1, 3, 4, 6} and 5 comes up twice as often as {1, 3, 4, 6}. Answer the following questions about this die. (a) Find the probability distribution p for this loaded die. (b) What is the probability that you roll an odd number with the die? (c) What is the probability that you roll the die twice and the two rolls sum to 7? (d) What is the probability that you roll the die twice and the two rolls sum to 7 given that the first roll is an even number? (e) Based on your answers to parts (c) and (d), are the events that you roll an even on the first roll and the two dice sum to 7 independent?

Explanation / Answer

a)Probability of getting any of these {1,3,4,6} = 1/9

Probability of getting {2} = 3/9

Probability of getting {5} = 2/9

b) P(odd number on die) = P(1)+P(3)+P(5) = 1/9 + 1/9 + 2/9 = 4/9

c) P (rolled twice and sum 7 ) = P (1 and 6) + P (2 and 5) + P (3 and 4) + P (4 and 3) + P(5 and 2) + P(6 and 1)

= 2[ P (1 and 6) + P (2 and 5) + P (3 and 4) ]

= 2{1/9*1/9 + 2/9 * 3/9 + 1/9*1/9} = 2/81 {1 + 6 + 1} = 16/81

d) P(Sum 7 with first number even) = P (6,1) + P(2,5) + P(4,3) = 1/9*1/9 + 2/9 * 3/9 + 1/9*1/9 = 8/81

P(first even number ) = 1- P(odd) =1-4/9 = 5/9

or P (sum 7 with first number even /even first )= (8/81)/(5/9) = 1/5

e) from c and d ,No events are not independent

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