A coin is tossed three times. Determine which of the following outcomes describe mutually exclusive events.

a) A: all tosses are heads , B: all tosses are tails
b) A: at least one toss is a tail , B: at least one toss is a head
c) For both a) and b) find P(A or B) and P( A and B)
d) Which pairs of events A and B, if any, as described in parts a) and b) are independent events?

To determine which of the outcomes describe mutually exclusive events, we need to understand what mutually exclusive events mean.

Mutually exclusive events are events that cannot occur at the same time. In other words, if event A happens, event B cannot happen, and vice versa.

Let's analyze each option:

a) A: all tosses are heads , B: all tosses are tails
In this case, events A and B are mutually exclusive because it is not possible for all tosses to result in heads and tails at the same time.

b) A: at least one toss is a tail , B: at least one toss is a head
In this case, events A and B are not mutually exclusive because it is possible to have at least one tail and at least one head in the three tosses.

Next, let's calculate the probabilities:

c) For both a) and b) find P(A or B) and P( A and B):

For option a):
P(A or B) can be calculated by adding the probability of event A happening (all tosses are heads) to the probability of event B happening (all tosses are tails).
P(A or B) = P(A) + P(B) = (1/2)^3 + (1/2)^3 = (1/8) + (1/8) = 1/4

P(A and B) is zero because events A (all tosses are heads) and B (all tosses are tails) cannot happen simultaneously.

For option b):
P(A or B) can be calculated by subtracting the probability of the complement of both events' outcomes from 1.
P(A or B) = 1 - P(neither A nor B) = 1 - P(all tosses are heads) - P(all tosses are tails) = 1 - (1/2)^3 - (1/2)^3 = 1 - 1/8 - 1/8 = 3/4

P(A and B) can be calculated by multiplying the probability of event A happening (at least one toss is a tail) by the probability of event B happening (at least one toss is a head).
P(A and B) = P(A) * P(B) = 1 - P(neither A nor B) = 1 - (1/2)^3 - (1/2)^3 = 1 - 1/8 - 1/8 = 3/8

d) To determine whether events A and B are independent, we need to compare the probabilities. If P(A and B) = P(A) * P(B), then events A and B are independent.

For option a), the events A and B are mutually exclusive; therefore, they cannot be independent.

For option b), the events A and B are not mutually exclusive. To check for independence, we compare P(A and B) to P(A) * P(B). If P(A and B) = 3/8, and P(A) * P(B) = (3/4) * (3/4) = 9/16, it can be concluded that events A and B are not independent.

In summary:
a) Events A and B are mutually exclusive.
b) Events A and B are not mutually exclusive.
c) P(A or B) for a) is 1/4; P(A and B) is 0.
P(A or B) for b) is 3/4; P(A and B) is 3/8.
d) Neither pair of events A and B from options a) and b) are independent.