An unbiased coin is tossed three times. If A is the event that a head appears on each of the first two tosses,B is the event that a tail occurs on the third toss and C is the event that exactly two tails appear in the three tosses,show that:
i) Events A and B are independent.
ii) Events B and C are dependent.
To show that events A and B are independent, we need to demonstrate that the probability of both events occurring is equal to the product of their individual probabilities.
Event A: A head appears on each of the first two tosses.
The probability of a head appearing on a single coin toss is 1/2, assuming the coin is unbiased.
Since two coins are tossed independently, the probability of a head appearing on each of the first two tosses is (1/2) * (1/2) = 1/4.
Event B: A tail occurs on the third toss.
The probability of a tail appearing on a single coin toss is 1/2.
Therefore, the probability of a tail occurring on the third toss is also 1/2.
To show that events A and B are independent, we need to compare their joint probability with the product of their individual probabilities:
P(A and B) = P(A) * P(B)
P(A and B) = (1/4) * (1/2)
P(A and B) = 1/8
Since the joint probability P(A and B) is equal to the product of the individual probabilities P(A) and P(B), we can conclude that events A and B are indeed independent.
Moving on to events B and C, we will show that they are dependent. To do this, we need to demonstrate that the probability of both events occurring is different from the product of their individual probabilities.
Event B: A tail occurs on the third toss.
The probability of a tail appearing on a single coin toss is 1/2.
The probability of getting exactly two tails in three tosses (event C) depends on the specific outcomes of the first two tosses.
Let's consider the following scenarios for the first two tosses:
1) HT (head on the first toss, tail on the second)
2) TH (tail on the first toss, head on the second)
3) TT (tail on both tosses)
If the first two tosses result in a tail (TT), event B (tail on the third toss) is guaranteed. Therefore, the probability of events B and C occurring together is 1/2.
But if the first two tosses result in HT or TH, the probability of getting exactly two tails in the three tosses (event C) would be 1/4, as there is only one favorable outcome (TT) out of the four possible outcomes (HHH, HHT, HTH, and HTT in the case of HT as the first two tosses, and TTH, THT, THH, and TTT in the case of TH as the first two tosses). In these cases, events B and C are not guaranteed to occur together, resulting in a probability of less than 1/2.
Since the probability of events B and C occurring together is different depending on the outcomes of the first two tosses, we can conclude that events B and C are dependent.