Question

1. By considering events concerned with independent tosses of a red die and a blue die, or otherwise. Give examples of events A, B, and C which are not independent, but nevertheless are such that every pair of them is independent.

2. By considering events concerned with three independent tosses of a coin and supposing that A and B both represent tossing a head on the first trial, give examples of events A, B and C which are such that P(ABC) = P(A)P(B)P(C) although no pair of them is independent.

Answer 1

1)

We throw two dices, one is red and the other is blue.

Let A be the event “the sum of the points is 7”, B the event “die red came up 3”, and C the event “die blue came up 4”.

Now, .

Also,

so that all events are pairwise independent.

However,

while

so they are not independent as a triplet.

2)

Let

A= At least two head

B= Head in the last coin

C= Head in the first coin

So

and

and you can check that these events are not pair-wise independent as it is checked is the previous example.

Thanks