Consider an urn containing ten balls; one of them with the number 1 on it, two o
ID: 3306136 • Letter: C
Question
Consider an urn containing ten balls; one of them with the number 1 on it, two of them with the number 2 on them, three of them with the number 3 on them, and four of them with the number 4 on them. Suppose that a ball will be randomly selected from the ten balls in the urn, and the number on it observed, and then a second ball will be randomly drawn from the nine remaining balls.
(a) What is the probability that the number on the 2nd ball will be greater than the number on the first ball?
(b) What is the probability that the number on the 1st ball will be a 2, given that the number on the 2nd ball will be greater than the number on the first ball?
(c) Letting A be the event that the number on the first ball will be a 1, and B be the event that the number on the 2nd ball will be a 2, are A and B independent events? Answer yes or no, and justify your answer using the definition of two events being independent.
Explanation / Answer
Total number of outcomes for selecting 2 balls out of 10 without replacement os 10*9 = 90
a. Favourable outcomes along with the number of ways
(1.2) - 1*2 =2
(1,3)- 1*3=3
(1,3)- 1*3=3
(1,4)-1*4=4
(2,3)-2*3=6
(2,4)-2*4=8
(3,4)-3*4=12
Total -35
Prob = 35/90=0.38
b.
Total number on 2nd ball will be greater than the number on the first ball = 35
Total number of 1st ball number being 2
6+8 =14
P = 14/35=0.4
c.
No
P(A) = 1/10
P(B)
It has 2 case
Total is i.+ii.= 2/9
And
When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring. Hence Event B is affected by event A as it is without replacement and the count of the total ball reduces.
So, for Independent Events:
P(A and B) = P(A) × P(B)
P(A and B) = 2/90 and P(a) *P(B) = 3/90
Hence, both are interdependent