A coin is tossed 3 times. Find the probability that all the 3 are heads
(A) if it is known that the first is heads
(B) if it is known that the first 2 are heads
(C) if it is known that the 2 of them are heads
To find the probability in each scenario, we first need to understand the total number of possible outcomes when tossing a coin 3 times. Since each coin toss has 2 possible outcomes (heads or tails), the total number of outcomes for 3 coin tosses is 2^3 = 8.
Now, let's calculate the probability for each scenario:
A. If it is known that the first coin toss is heads:
In this case, we only need to consider the remaining 2 coin tosses. Since the first coin toss is fixed, there are 2^2 = 4 possible outcomes for the remaining 2 coin tosses. Out of these 4 outcomes, only one of them has all heads (HH). Therefore, the probability is 1/4.
B. If it is known that the first 2 coin tosses are heads:
Similar to scenario A, the first 2 coin tosses are fixed, and we only need to consider the remaining toss. So, there are 2 possible outcomes for the last coin toss (heads or tails). Only one of these outcomes will result in all 3 coin tosses being heads (HHH). Therefore, the probability is 1/2.
C. If it is known that 2 of the coin tosses are heads:
In this scenario, we already have 2 heads, so only 1 more coin toss is left. Hence, there are 2 possible outcomes for the last coin toss (heads or tails). Only one of these outcomes will result in all 3 coin tosses being heads (HHH). Therefore, the probability is 1/2.
To summarize:
A. The probability that all 3 coin tosses are heads, given that the first is heads, is 1/4.
B. The probability that all 3 coin tosses are heads, given that the first 2 are heads, is 1/2.
C. The probability that all 3 coin tosses are heads, given that 2 of them are heads, is 1/2.