Two numbers are selected at random (without replacement) from first 7 natural numbers. If X denotes the smaller of the two numbers obtained, find the probability distribution of X. Also, find mean of the distribution.

OR

There are three coins, one is a two headed coin (having head on both the faces), another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. If It shows head. What is probability that it was the two headed coin?

Given: Two numbers are selected at random (without replacement) from first 7 natural numbers

To Find: Find the probability distribution of smaller of the two numbers obstained. Also, find mean of the distribution

Total sample space={(1,2),(1,3),(1,4),(1,5),(1,6),(1,7), (2,1),(2,3),(2,4),(2,5),(2,6),(2,7),(3,1),(3,2),(3,4),(3,5),(3,6),(3,7),(4,1),(4,2),(4,3),(4,5),(4,6),(4,7),(5,1),(5,2),(5,3),(5,4),(5,6),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5),(6,7),(7,1),(7,2),(7,3),(7,4),(7,5),(7,6)}

Total possible states = 42

Among first 7 numbers, the smaller numbers can be 1, 2, 3, 4, 5, 6.

States containing 1 as smaller number = 12

States containing 2 as smaller number = 10

States containing 3 as smaller number = 8

States containing 4 as smaller number = 6

States containing 5 as smaller number = 4

States containing 6 as smaller number = 2

Probability distribution of X is,

Mean =

=

OR

Given: There are three coins, one is a two headed coin (having head on both the faces), another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. It shows head

To Find: Probability that it was the two headed coin

Tip: Use Bayes’ Theorem

Let, P₁ is the probability of choosing two headed coin.

P₂ is the probability of choosing biased coin.

P₁ is the probability of choosing non-biased coin.

And also, X be the probability of appearing head.

So,