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 appears in the three tosses, show that :

i) Event A and B are independent
ii) Event B and C are dependent

To show whether two events are independent or dependent, we need to compare the probabilities of the two events occurring separately vs. occurring together.

i) To show that events A and B are independent, we need to show that the probability of both events occurring together 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 occurring on each of the first two tosses is 1/2 * 1/2 = 1/4, since the coin is unbiased.

Event B: A tail occurs on the third toss.
The probability of a tail occurring on the third toss is 1/2, since the coin is unbiased.

Now, let's find the probability of both events A and B occurring together:
- There are 2 possible outcomes for the first two tosses (HH, HT).
- The probability of getting a tail on the third toss is 1/2.
- The probability of getting heads on the first two tosses is 1/4.

Since the coin tosses are independent, we can multiply the probabilities:
P(A and B) = P(A) * P(B) = (1/4) * (1/2) = 1/8.

Therefore, events A and B are independent since the probability of both occurring together is equal to the product of their individual probabilities.

ii) To show that events B and C are dependent, we need to compare the probability of event B occurring with and without event C.

Event B: A tail occurs on the third toss.
The probability of a tail occurring on the third toss is 1/2, since the coin is unbiased.

Event C: Exactly two tails appear in the three tosses.
The probability of exactly two tails occurring can be calculated using combinations. Out of the 3 tosses, there are 3 possible combinations with two tails: TTH, THT, HTT. Each combination has a probability of (1/2)^2 * (1/2) = 1/8.

Now, let's find the probability of event B occurring without event C (B | not C):
- The probability of event B occurring when not C means that both heads appeared in the first two tosses.
- There are 2 possible outcomes (HH, HT) where both heads appeared in the first two tosses.
- The probability of event B (tail on the third toss) when not C is 0, since there are no tails in those outcomes.

Since the probability of event B occurring (1/2) is not equal to the probability of event B occurring when not C (0), events B and C are dependent.

Therefore, we have shown that event A and B are independent (i), while event B and C are dependent (ii).