We will answer the following questions using the Prob Library in R:
1) A dice is rolled once. Find the probability of obtaining
a) the number 3
b) an odd number
c) a number greater than 2
Solution
a) P(3) = 0.17
R Code:
> require(prob)
> S<-rolldie(1,makespace = T)
> A<-subset(S,X1==3)
> Prob(A)
[1] 0.1666667
> round(Prob(A),digits=2) # the function round() rounds off the probability to 2 decimal places.
[1] 0.17
b) P(X is an odd number) = P(X = 1 or X = 3 or X = 5) = 0.5
R Code:
> S<-rolldie(1,makespace = T)
> B <-subset(S,X1==1 | X1==3 | X1==5)
> Prob(B)
[1] 0.5
c) P(X > 2) = 0.67
R Code:
> S<-rolldie(1,makespace = T)
> C <-subset(S,X1>2)
> Prob(C)
[1] 0.6666667
> round(Prob(C), digits=2)
[1] 0.67
2) Two dice are rolled together. Find the probability of obtaining
a) two different digits
b) a difference of 1
c) a sum of 10
Solution
a) P(X1 is not equal to X2) = 0.83
R Code:
> S<-rolldie(2,makespace = T)
> D <-subset(S,X1!=X2)
> D
X1 X2 probs
2 2 1 0.02777778
3 3 1 0.02777778
4 4 1 0.02777778
5 5 1 0.02777778
6 6 1 0.02777778
7 1 2 0.02777778
9 3 2 0.02777778
10 4 2 0.02777778
11 5 2 0.02777778
12 6 2 0.02777778
13 1 3 0.02777778
14 2 3 0.02777778
16 4 3 0.02777778
17 5 3 0.02777778
18 6 3 0.02777778
19 1 4 0.02777778
20 2 4 0.02777778
21 3 4 0.02777778
23 5 4 0.02777778
24 6 4 0.02777778
25 1 5 0.02777778
26 2 5 0.02777778
27 3 5 0.02777778
28 4 5 0.02777778
30 6 5 0.02777778
31 1 6 0.02777778
32 2 6 0.02777778
33 3 6 0.02777778
34 4 6 0.02777778
35 5 6 0.02777778
> Prob(D)
[1] 0.8333333
>round(Prob(D),digits=2)
[1] 0.83
b) P(|X1 - X2 |= 1) = 0.28
R Code:
> E <-subset(S,abs(X1-X2)==1)
> E
X1 X2 probs
2 2 1 0.02777778
7 1 2 0.02777778
9 3 2 0.02777778
14 2 3 0.02777778
16 4 3 0.02777778
21 3 4 0.02777778
23 5 4 0.02777778
28 4 5 0.02777778
30 6 5 0.02777778
35 5 6 0.02777778
> Prob(E)
[1] 0.2777778
> round(Prob(E),digits = 2)
[1] 0.28
c) P(X1 + X2 = 10) = 0.08
R Code:
> F <-subset(S,X1+X2==10)
> F
X1 X2 probs
24 6 4 0.02777778
29 5 5 0.02777778
34 4 6 0.02777778
> Prob(F)
[1] 0.08333333
> round(Prob(F),digits = 2)
[1] 0.08
3) What is the conditional probability that a die lands on a prime number, given that it lands on an odd number?
P(Prime given Odd) = 0.67
R Code:
> S<-rolldie(1,makespace = T)
> A <-subset(S,X1==2 | X1==3 | X1==5)
> B <-subset(S,X1==1 | X1==3 | X1==5)
> Prob(A, given=B)
[1] 0.6666667
> round(Prob(A, given=B),digits=2)
[1] 0.67
Alternatively, we can solve this probability as follows.
> S<-rolldie(1,makespace=T)
> A <-subset(S,X1==2|X1==3|X1==5)
> B <-subset(S,X1==1|X1==3|X1==5)
> Prob(intersect(A,B))/Prob(B)
[1] 0.6666667
> round(Prob(intersect(A,B))/Prob(B),digits=2)
[1] 0.67