A high school basketball team leads at halftime in 60% of the games in a season. The team wins 80% of the time when it has the halftime lead, but only 10% of time when they do not. We want to find the probability that the team wins a particular game during the season. Let's start by naming the occurring events.
A& - Team leads at halftime.
B& - Team wins.
Note that when event A occurs P(B)=80%=0.8, and when event A does not occur P(B)=10%=0.1. The probability of event B depends on the occurrence of event A, so the events are dependent. Let's now consider the probabilities on a probability tree diagram. Note that the event of the team losing is a complement of event A, so we will mark it as A.
Since on the tree diagram there are complements of events A and B, we can use the formula for the probability of a complement to calculate their probabilities.
P(A)& =1- 0.6= 0.4
P(B|A)& =1- 0.8= 0.2
P(B|A)& =1- 0.1= 0.9
Let's add the obtained information to our diagram.
To find the probability that the team wins a particular game we need to follow the branches of our diagram leading to event B. There are two ways leading to event B. We have to multiply the values on each way, and then add the values from both ways.
