There are five balls labelled 1, 2, 3, 4, and 5 in a box. Without looking, you w

Question

There are five balls labelled 1, 2, 3, 4, and 5 in a box. Without looking, you will take out two balls (without replacement).

a) If we are not interested in the order, how many outcomes are there?

b) Let a random variable, X, denote the maximum of the two numbers. Create a proper PMF table for the random variable X.

c) What is the probability that the maximum of the two numbers is at least 4?

d) Given that the maximum of the two numbers is less than 5, what is the probability that it is at most 3?

Explanation / Answer

a)

number of outcomes =N(Selecting 2 balls out of 5)=5C2 =10

b)

below is PMF table for the random variable X

P(X=1) =0 (as there is only 1 ball numbered 1 ; and second must be greater than this)

P(X=2)=1/10 (only one event (1.2) where 2 can be maximum)

P(X=3)=2/10

P(X=4)=3/10

P(X=5)=4/10

c)

probability that the maximum of the two numbers is at least 4 =P(X=4)+P(X=5)=(3/10)+(4/10)=7/10

d)

probability that it is at most 3 given  less than 5 =P(X<=3|X<5)

=(P(X=1)+P(X=2)+P(X=3))/(P(X=1)+P(X=2)+P(X=3)+P(X=4))

=(0/10+1/10+2/10)/(0/10+1/10+2/10+3/10)=(3/10)/(6/10)=3/6 =1/2

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