## Saturday, July 28, 2012

Independent Events

Events are considered independent events if the occurrence of one has no bearing on the probability of the occurrence of the other. A good example to illustrate this is the tossing of a fair coin. A coin has no memory, meaning that even if a head comes up 10 times in a row, the probability that the next toss is a head is still ½. The probability that the next toss is a tail is also ½. The probability that the toss of a fair coin lands heads up or tails up on any single toss is always ½ no matter how many times the coin has been previously tossed.

If two events A and B are independent, then P(A B) = P(A) ∙ P(B).

Note that the rule for independent events can be extended for more than 2 events. If there are 3 independent events, A, B and C, then P(A B ∩ C) = P(A) ∙ P(B) ∙ P(C). Similar results are seen for 4 events, 5 events and so on.

Example: Suppose a spinner has 12 regions numbered 1 – 12 each of which the spinner is equally likely to land on. What is the probability that the spinner will land on 2, 2, 10, 9 and any odd number in 5 consecutive spins?

P(2) = 1/12

P(10) = 1/12

P(9) = 1/12

P(odd) = ½

Therefore, since these are independent events, the probability the spinner lands on 2, 2, 10, 9 and any odd number in 5 consecutive spins is

(1/12)(1/12)(1/12)(1/12)(1/2) = 1/41,472