we have 2 coins: coin 1 comes up with the probability of tails of 2/3 while coin 2 comes up with the probability of tails of 1/3 you don't know which is which so you pick one up at random and flip it . If its tails, you guess that the coin you picked up is coin 1 else you guess that the coin you picked up is coin 2. what is the probability that your guess is correct.
To solve this problem, we can use Bayes' theorem.
Let's define the events:
A = Coin 1 is chosen
B = Coin 2 is chosen
T = Tails is observed
We want to find the probability that your guess (your choice of coin) is correct. In other words, we want to find P(A|T), the probability that you picked coin 1 given that tails is observed.
According to Bayes' theorem:
P(A|T) = (P(T|A) * P(A)) / P(T)
Given the information provided, we know:
P(T|A) = 2/3 (probability of tails given coin 1 is chosen)
P(T|B) = 1/3 (probability of tails given coin 2 is chosen)
P(A) = P(B) = 1/2 (probability of choosing either coin)
Now, let's calculate P(T), the probability of observing tails:
P(T) = P(T|A) * P(A) + P(T|B) * P(B)
= (2/3) * (1/2) + (1/3) * (1/2)
= 2/6 + 1/6
= 3/6
= 1/2
Now, substitute the values into Bayes' theorem:
P(A|T) = (P(T|A) * P(A)) / P(T)
= (2/3 * 1/2) / (1/2)
= 2/6 / 1/2
= 2/6 * 2/1
= 4/6
= 2/3
Therefore, based on the given information and your method of guessing, the probability that your guess is correct is 2/3.