Final answer:
The probability that if you reflip the coin it will come up heads again is 20%.
Step-by-step explanation:
To determine the probability that if you reflip the coin it will come up heads again, you need to consider the probabilities of each coin in the bag and the result of the first flip.
Let's define the events:
- A: The randomly selected coin is the one with a 60% chance of coming up heads.
- B: The first flip result is heads.
- C: The reflip of the coin comes up heads again.
We want to find P(C|B), the probability of C given B. Using Bayes' theorem, we have:
P(C|B) = P(B|C) * P(C) / P(B)
P(B|C) is the probability of getting heads on the reflip, given that the coin is the one with a 60% chance of coming up heads, which is 0.6.
P(C) is the overall probability of choosing the coin with a 60% chance of coming up heads, which is 1/3 since there are three visually indistinguishable coins in the bag.
P(B) is the overall probability of getting heads on the first flip, and it can be calculated using the law of total probability:
P(B) = P(B|A) * P(A) + P(B|B) * P(B) + P(B|C) * P(C)
Since the coins are visually indistinguishable, the probability of choosing each coin is the same, so P(A) = P(B) = P(C) = 1/3.
P(B|A) is the probability of getting heads on the first flip, given that the coin is the one with a 20% chance of coming up heads, which is 0.2.
P(B|B) is the probability of getting heads on the first flip, given that the coin is the one with a 20% chance of coming up heads, which is 0.2.
P(B|C) is the probability of getting heads on the first flip, given that the coin is the one with a 60% chance of coming up heads, which is 0.6.
Substituting the values into the equation, we have:
P(B) = 0.2 * (1/3) + 0.2 * (1/3) + 0.6 * (1/3) = 0.2.
Finally, we can calculate P(C|B):
P(C|B) = (0.6 * (1/3)) / 0.2 = 0.2.
Therefore, the probability that if you reflip the coin it will come up heads again is 0.2, or 20%.