3. (a) What is the probability that if three independent dice are tossed, none h
ID: 3048475 • Letter: 3
Question
3. (a) What is the probability that if three independent dice are tossed, none have a value of 1. Do not list the combinations. (b) A four-sided die has equally likely integer outcomes 1, 2, 3, and 4 (rather than 1 to 6). If three dice are tossed, what is the probability that we get a sum equal to or greater than 9. Show all combinations. P(sun? 9) = (c) Show all possible combinations of the tosses of five dice, resulting in a sum of 7. What is the probability of this event? Hint: All outcomes require either three or four 1's.Explanation / Answer
1) When a single dice rolls, the probability of getting 1 on the face = 1/6
As the three dice are independent, then the probability of getting any number is independent of the other dice or the probability of getting X is not affected by what number the other dice gets. It is purely independent.
Thus, for none of the three dice to get a 1, that means any number can roll on the dice i.e we can get 2,3,4,5,6 on the dice but NOT a 1.
Therefore, probability that none of the dice get a 1 = (Prob of not getting a 1) x (Prob of not getting a 1) x (Prob of not getting a 1)
= 5/6 x 5/6 x 5/6 = 125/216
Ans: Prob of getting no>
2) For three 4 sided die with numbers = (1,2,3,4) --->
Total number of outcomes = 4 x 4 x 4 outcomes = 64 outcomes
Outcomes that result in sum greater than or equal to 9 ---> (1,4,4) (2,3,4) (2,4,3) (2,4,4) (3,2,4) (3,3,3) (3,3,4) (3,4,2) (3,4,3) (3,4,4) (4,1,4) (4,2,3) (4,2,4) (4,3,2) (4,3,3) (4,3,4) (4,4,1) (4,4,2) (4,4,3) (4,4,4) = 20 outcomes
Ans: P( sum >= 9 ) = 20/64 = 5/16
3) Probability of sum on 5 dice = 7 ----->
Total outcomes = 6 x 6 x 6 x 6 x 6 = 7776 outcomes
Matching outcomes (Using the hint) --->
3 ones : If we have 3 ones, that means we have 2 two's. 1 + 1 + 1 + 2 + 2 = 7
4 ones : If we have 4 ones, that means we have 1 three. 1 + 1 + 1 + 1 + 3 = 7
a) 3 ones: We need to find the possible permutations of 1, 1, 1, 2, 2 ------>
1,1,1,2,2 ; 1,1,2,1,2 ; 1,1,2,2,1 ; 1,2,1,1,2 ; 1,2,1,2,1 ; 1,2,2,1,1 ; 2,1,1,1,2 ; 2,1,1,2,1 ; 2,1,2,1,1 ; 2,2,1,1,1
b) 4 ones: We need to find the possible permutations of 1, 1, 1, 1, 2 ------>
1,1,1,1,2 ; 1,1,1,2,1 ; 1,1,2,1,1 ; 1,2,1,1,1 ; 2,1,1,1,1
Matching outcomes = 10 + 5 = 15 outcomes
Ans: Prob(Sum on 5 dice = 7) = 15 / 7776 = 0.00193
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