Hello, math8553!

= normal coin, . = double-head coin.Suppose i have 3 coins in a bag: two ordinary coins, one with two heads.

I take a coin out of my bag at random.

Then: .

There are two possibilites:a) I toss the coin.

What is the probability it comes up Heads?

[1] The -coin is chosen and it comes up Heads: .

[2] The -coin is chosen and it comes up Heads: .

Therefore: .

Given that coin turned up Heads, what is the probability that it is the -coin?b) Suppose I take a coin randomly, toss it, and it comes up Heads.

What is the probability that the other side is a Head?

This is Conditional Probability which requires Bayes' Theorem: .

We want: .

c) )What is the probability that if i toss the same coin a second time i will get another head?

We want: .

There are two ways to get

[1] -coin and two Heads: .

[2] -coin and two Heads: .

. . Hence: .

Therefore: .