Question

Three fair dice are rolled. Let S be the total number of spots showing, that is the sum of the results of the three rolls.

a) Find the probabilities P(S = 3), P(S = 4), P(S = 17), P(S = 18).

b) Find the probability P(S ≥ 11).

Answer 1

Solution:

Given : Three dice are rolled and S denotes the sum obtained by rolling three dice.

Total number of outcomes = 6^3 = 216.

a)

1. P(S=3) means sum obtained is 3. (this is possible only when 1 occurs on all three dice)

Here, Sample space = 1 (1+1+1)

By the formula Prob=Favourable outcomes/ Total outcomes.

P(S=3) = 1/216 = 0.005%

2. P(S=4) means sum obtained is 4.

Here, Sample space = 1+2+1 , 2+1+1 , 1+ 1+2

Favourable outcomes = 3

By the formula Prob=Favourable outcomes/ Total outcomes.

P(S=4) = 3/216 = 0.014%

3. P(S=17) means sum obtained is 17.

Here, Sample space = 5+6+ 6, 6+ 5+6, 6+6+5  

Favourable outcomes = 3

By the formula Prob=Favourable outcomes/ Total outcomes.

P(S=17) = 3/216 = 0.014%

4. P(S=18) means sum obtained is 18. (this is possible only when 6 occurs on all three dice)

Here, Sample space = 1 (6+6+6)

P(S=18) = 1/216 = 0.005%

b) P(S>=11) means Sum=11, 12,13,14,15, 16,17,18

For P(S=11)

Sample Space=  6 + 4 + 1 (3! =6 ) ,1 + 5 + 5 (3) , 5 + 4 + 2 (6), 3 + 3 + 5 (3) , 4 + 3 + 4 (3) , 6 + 3 + 2 (6)

fav outcomes = 27

For P(S=12)

Sample Space=  6 + 5 + 1 (6) , 4 + 3 + 5 (6) ,4 + 4 + 4 (1), 5 + 2 + 5 (3) ,6 + 4 + 2 (6) 6 + 3 + 3 (3)

Fav outcomes = 25

For P(S=13)

Sample Space = 6 + 6 + 1 (3) ,5 + 4 + 4 (3), 3 + 4 + 6 (6) ,6 + 5 + 2 (6), 5 + 5 + 3 (3)

Fav outcomes = 21

For P(S=14)

Sample Space = 6 + 6 + 2 (3), 5 + 5 + 4 (3),  4 + 4 + 6 (3) ,6 + 5 + 3 (6)

Fav outcomes = 15

For P(S=15)

Sample Space = 6 + 6 + 3 (3) , 6 + 5 + 4 (6) , 5 + 5 + 5 (1)

Fav outcomes = 10

For P(S=16)

Sample Space = 6 + 6 + 4 (3) ,5 + 5 + 6 (3)

Fav outcomes = 6

P(S=17) Fav outcomes = 3 (explained above in (a))

P(S=18) Fav outcomes =1 (explained above in (a))

P(S>=11) = (27+25+21+15+10+6+3+1)/ 216 = 108/216 =0.5%