have three coins, two of which are fair and the other is a double header .suppose a coin is selected using random selection and tossed twice,one after the other.if i got two heads what is the probability that the coin that was selected was the double header?

To find the probability that the selected coin is the double header, given that you obtained two heads in two tosses, you can use Bayes' theorem. Let's break down the problem step by step:

1. Define the events:
- A: Coin selected is the double header
- B: Two heads obtained in two tosses

2. Assign probabilities:
- P(A) = Probability that the selected coin is the double header = 1/3 (since you have three coins and one of them is the double header)
- P(B) = Probability of getting two heads in two tosses

3. Determine the probabilities of events A and B given A has occurred:
- P(A|B) = Probability that the selected coin is the double header, given two heads obtained

4. Apply Bayes' theorem:
- Bayes' theorem states: P(A|B) = (P(B|A) * P(A)) / P(B)

To determine P(B|A), we need to calculate the probability of getting two heads with a double header:
- P(B|A) = Probability of obtaining two heads with a double header = 1 (since the double header always gives two heads in two tosses)

The probability of getting two heads in two tosses can be calculated by considering two scenarios:
- Scenario 1: The double header is selected and two heads are obtained (P(B|A) * P(A))
- Scenario 2: One of the fair coins is selected and two heads are obtained (P(B|¬A) * P(¬A)), where ¬A represents the complement of event A

So, P(B) = (P(B|A) * P(A)) + (P(B|¬A) * P(¬A))

5. Calculate the probabilities and substitute them into Bayes' theorem to find P(A|B):
- P(A|B) = (P(B|A) * P(A)) / [(P(B|A) * P(A)) + (P(B|¬A) * P(¬A))]

Once you have substituted the appropriate values into the equation, you can calculate the probability that the selected coin was the double header, given that two heads were obtained in two tosses.