Probabilities II: Exclusive and independent events

Two events are exclusive if they cannot happen at the same time. If two events A and B are exclusive, then P(A or B) = P(A) + P(B).

For example, in the experiment of throwing a die, the events "getting an even number" and "getting a '1'" are exclusive, because they cannot occur at the same time: the result cannot be an even number (2, 4 or 6) and AT THE SAME TIME, 1.

Here, if
A = {getting an even number}
B = {getting a '1'}

and we want to find the probability of A or B occurring, then

P(A or B) = P(A) + P(B) = 3/6 + 1/6 = 4/6 = 2/3.


Two events are independent if the occurrence of one doesn't depend on the occurrence of the other. If two events A and B are independent, then P(A and B) = P(A) . P(B)

For example, in the experiment of tossing a coin two times, we can get two possible results (head or tail), then n=2.

If
A = {getting a 'head' in the first toss}
B = {getting a 'tail' in the second toss}

then, A and B are independent (because the event of getting a 'tail' in the second toss is unaffected by the event of getting a 'head' in the first one; they don't relate to each other).

Then, the probability of getting a 'head' in the first toss AND a 'tail' in the second toss is P(A and B) = P(A).P(B) = 1/2 . 1/2 = 1/4.


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PRACTICE FROM THE BOOK: Page 354, Exercise 9.