Question

If you roll two six-sided dice, what is the probability of obtaining the following outcomes?

a)2 or 3

b) 6 and 4

c) At least one 5

d) Two of the same number (two 1s, or two 2s, or two 3s, etc.)

e) An even number on both dice

f) An even number on at least one die

Answer 1

If we rolled 2 six-sided dice,

a) the probability of obtaining 2 or 3 is = 22/36 = 11/18

2 dice are rolled. therefore there are total 36 possibilities. and among this 36 possibilities no of times we can obtain 2 is 11. those are (2,1),(2,2),....,(2,6)(1,2),(3,2),.....,(6,2) .similarly no of times we can obtain 3 is 11. probability of getting either 2 or 3 is 22/36.

b) the probability of obtaining 6 and 4 is 2/36 = 1/18

among 36 possibilities 6 and 4 can be obtained in 2 cases (6,4) and (4,6). so the probability is 2/36=1/18.

c) probability of getting at least one 5 is 11/36

no. of cases where at least one 5 will occur (1,5),(2,5),...(6,5),(5,1),...,(5,4),(5,6) =11

d)Probability of obtaining two same nos is: 1/6

sample space of two same no's are {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}.so there are 6 combinations. probability = 6/36 =1/36

e)Probability of getting an even no on both dice are 1/4

A die can takes value {2,4,6} i.e all possible cases are {(2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2)(6,4),(6,6)}. total no of cases are 9. probability = 9/36=1/4

f)Probability of getting an even no on at least on die is 11/12

all possible cases where 2 can occur is 11. 4 can occur in 11 place similarly 6 can occur in 11 place.

i.e. probability = (11+11+11)/36 = 33/36 = 11/12