Question

Problem 2. (30 points) a) (5 points) In rolling 3 fair dice, what is the probability of obtaining a sum not greater than 77 b) (5 points) In rolling 2 fair dice, what is the probability of a sum greater than 3 but not exceeding 6? o) (5 points) Given that the frstroll was an odd number what is the probability that sum exceeds 6? The notation for this is P(A I B)- Probability(sum exceeds 6 given that the first roll was 1, 3 or 5). d) (5 points) The definition for independent events: two events, A and B, are independent if P(A | B) - P(A). Given events A - sum exceeds 6, and B - first roll was an odd number,'are they independent? In other words, (one of the student's words): Given that the first die was an odd number (1, 3, 5), what is the probability that the sum exceeds 6 when rolling the second die. e) (5 points) what if events A-both rolls are the same, and B # first roll was an odd number? f) (5 points) Are the results from d and e surprising? Interpret.

Answer 1

(a)

Total number of outcomes when three dice are rolled is 6*6*6 = 216

The outcomes of three dice that has sum 7 or less are:

One outcomes having: 1 , 1 ,1 , sum =3

Three outcomes having: 1, 1, 2 , sum = 4

Three outcomes having: 1, 1, 3, sum = 5

Three outcomes having: 1,1,4, sum = 6

One outcome having: 2,2,2, sum =6

Three outcomes having: 2,2,1, sum =5

Six outcomes having: 1,3,2, sum = 6

Three outcomes having: 1,3,3, sum = 7

Six outcomes having: 1,4,2, sum = 7

Three outcomes having: 1,1,5, sum = 7

Three outcomes having: 2,2,3, sum = 7

Number of outcomes having sum less than equal to 7: 1+3+3+3+1+3+6+3+6+3+3 =35

The required probability is

P(sum is less than equal to 7) = 35 / 216 = 0.1620

(b)

Following is the possible outcomes when we sum the outocmes of two rolls of die (X):

(2.1)-3 (2,2)-4 (2. 3)-5 (2. 4)-6 (2,5)-7 (2,6)=8 (3.1): 4 (3,2)-5 (3, 3-6 (3,4) 7 (3,5)-. 8 (3,6)-9 (4,1) 5 4,2) 6 4,3) 7 (4,4) 8 4,5) 9 (4,6) 10 (5,1) 6 (5,2)7 (5,3) 8 (5,4)9 (5,5) 10 5,6) 11 (6,1)=7 (6,2)=8 (6. 3)=9 (6,4)=10 (6,5)=11 (6,6)=12

Out of 36 outcomes, 11 have sum greater than 3 and less than equal to 6 so

P( sum greater than 3 and less than equal to 6 ) = 11/36

(c)

P(sum is 6 | first roll is 1, 3, or 5) = 3/36 = 1/12

(d)

Out of 36 outcomes, 21 outcomes are greater than 6 so

P(sum is 6) = 21/36 = 7/12

Since P(sum is 6 | first roll is 1, 3, or 5) is not equal to P(sum is 6) so these are not independent.

(e)

P(A) = 6/36 = 1/6

P(B) = 18/36 = 1/2