A fair decahedral die has faces numbered 0-9, each equally likely to occur when

Question

A fair decahedral die has faces numbered 0-9, each equally likely to occur when the die is rolled. Round each answer to 4 places after the decimal point. A. If two fair decahedral dice are rolled, what is the probability that the sum of the numbers rolled is at most 4? B. If three fair decahedral dice are rolled, what is the probability that the sum of the numbers rolled is 14? C. If eight fair decahedral dice are rolled, what is the probability that at least one of the numbers rolled is 6? D. If a fair decahedral die is rolled 6 times, what is the probability that all the rolls result in a different number? E. If a fair decahedral die is rolled 9 times, what is the probability that only the last roll is at least 7?

Explanation / Answer

a)

Probability(P) = Favourable Outcomes (F) / Total Outcomes (T)

Two fair decahedral dice:

All possible combinations: {(0,0),(0,1),(0,2)…………………………………….. (9,9)} ie 10*10 = 100 combinations.

T = 100

Favourable Outcomes ie sum is at most 4 ie 0,1,2,3,4:

{(0,0),(0,1),(0,2),(0,4),(1,0),(1,1),(1,2),(1,3),(2,0),(2,1),(2,2),(3,0),(3,1),(4,0)} ie 15 combinations

F = 15

Hence,

P = 15/100 = 0.15

b)

Three fair decahedral dice:

T = 10*10*10 = 1000

Sum = 14

ie x1 + x2+x3 = 14

X1 can take any value (0,1,2,3,4)

X2 (1,2,3,4,5,6,7)

X3 ( 5,6,7,8,9)

Any combination of x1,x2,x3 is taken from above values such that their sum is 14

Eg:

(0,5,9),(0,6,8), (0,7,7) etc

There will be two types , one with repetition(0,7,7) and one’s without repetition (0,5,9).

Repetition combinations:

(0,7,7),(2,6,6),(3,3,8),(4,4,6),(4,5,5) – Each can be arranged into 3 ways. Hence total combinations – 15

Without Repetition combinations:

(0,5,9), (0,6,8),(1,3,9),(1,5,8),(1,6,7),(2,3,9),(2,4,8),(2,5,7),(3,4,7),(3,5,6),(4,1,9),(4,2,8) – Each can be arranged into 6 ways . Hence total combinations – 72

Hence,

F = 15 + 72 = 87

So,

P = 87/1000 = 0.087

d)

fair decahedral dice is rolled 6 times or 6 dice are rolled 1 times each.

T = 10^6

All the numbers are different:

F = 10*9*8*7*6*5

Hence, P = F/T = 0.1512

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