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 of getting all 3 heads in a coin toss, we need to determine the number of successful outcomes (getting all heads) over the total number of possible outcomes.
Let's start with the total number of possible outcomes when a coin is tossed 3 times. Since each coin flip has two possibilities (heads or tails), we can calculate it as 2^3 = 8.
(A) If it is known that the first toss is heads:
Since the first toss is already heads, we have one specific outcome in this case. So the number of successful outcomes is 1, and the probability would be 1/8.
(B) If it is known that the first 2 tosses are heads:
Similar to the previous case, we now know the first two tosses are heads. Again, we have only one specific outcome (HHH) in this case. Therefore, the number of successful outcomes is 1, and the probability would remain 1/8.
(C) If it is known that 2 of the tosses are heads:
In this case, we have to consider different combinations of three coin flips where exactly 2 of them are heads. Since there are three possible positions for the tails, we have three different outcomes (HHT, HTH, THH). Therefore, the number of successful outcomes is 3, and the probability would be 3/8.
So, the probabilities are:
(A) 1/8 or 0.125
(B) 1/8 or 0.125
(C) 3/8 or 0.375
To answer these questions, we need to apply the concept of conditional probability. We'll calculate the probability for each case:
(A) If it is known that the first coin flip is heads:
When we know that the first coin flip is heads, we only need to consider the remaining two coin flips. The probability of getting two more heads is (1/2) * (1/2) = 1/4. Thus, the probability that all three coins will be heads is 1/4.
(B) If it is known that the first two coin flips are heads:
When we know that the first two coin flips are heads, we only need to consider the last coin flip. The probability of getting a heads on the third flip is 1/2. Hence, the probability for all three coin flips being heads is 1/2.
(C) If it is known that two of the three coin flips are heads:
When we know that two of the three coin flips are heads, we need to consider the different possible scenarios:
1. The first two coin flips are heads and the last flip is tails: (1/2) * (1/2) * (1/2) = 1/8
2. The last two coin flips are heads and the first flip is tails: (1/2) * (1/2) * (1/2) = 1/8
3. The first and third coin flips are heads and the second flip is tails: (1/2) * (1/2) * (1/2) = 1/8
Considering these three scenarios, the probability that two of the three coin flips are heads is (1/8) + (1/8) + (1/8) = 3/8.
In summary:
(A) The probability that all three coin flips are heads if the first flip is heads is 1/4.
(B) The probability that all three coin flips are heads if the first two flips are heads is 1/2.
(C) The probability that all three coin flips are heads if two of the flips are heads is 3/8.