## Saturday, December 7, 2013

### Probabilities for independent events -- a run scoring example.

Question:  The probability that the Atlanta Braves score two or fewer runs on a particular night is 0.222.  The probability that the New York Yankees score two or fewer runs on a particular night is 0.290.

What is the probability that both teams score two or fewer runs on a given night?

What is the probability that either team scores two or fewer runs on a given night?

What is the probability that neither team scores two or fewer runs on a given night?

The probability of both events occurring is the probability of the intersection of the two events.  The events are independent; hence the probability of the intersection is the product of the two probabilities.

The probability of either event occurring is the probability of the union of two events, which is the sum of the probability of the two events minus the probability of the intersection of the two events.

The probability that both teams score two or fewer runs is( 0.222 x 0.290) or 0.0644.

The probability that either team scores two or fewer runs is (0.222 + 0.290 - 0.064) or 0.4476.

Neither team scoring two or fewer runs is the same as both teams scoring more than two runs.  The  probability of this is (1-0.222) x (1-0.290) or 0.5524.

Note that the probability that neither team gets more than two runs is the complement of the probability that either team gets more than two runs.   The probability of the complement of an event is 1 minus the probability of the event.  This means

P(neither team get two or more run) = 1 - P(either team gets two or more runs)

or

0.5524 = 1 - 0.4476

The run totals of the two teams are independent.   There are four outcomes from the two games (ATL<=2 & NY<=2), (ATL<=2, NY>2), (ATL>2, NY<=2) and (ATL>2, NY>2).  The probabilities of these four outcomes are presented in the table below.

 NY <= 2 NY>2 Atlanta <=2 0.06438 0.15762 Atlanta >2 0.22562 0.55238