Note that we are interested in the probability of two independent events. What formula do we need to use?
C
c
Compare the two obtained probabilities. Which situation has a higher probability of the Redbirds scoring and the Bluebirds not scoring?
A
a
9%
B
b
12%
C
c
The coach should take the goalie out.
Practice makes perfect
a
The Redbirds trail the Bluebirds by one goal with 1 minute left in a hockey game. The Redbirds' coach must decide whether to remove the goalie and add a frontline player. We are given the table showing the probabilities of each team scoring. We want to find the probability that the Redbirds score and the Bluebirds do not score when the coach leaves the goalie in.
Goalie
No Goalie
Redbirds Score
0.1
0.3
Bluebirds Score
0.1
0.6
Let's mark the event of the Redbirds scoring as R, the event of the Bluebirds scoring as B, and the event of leaving the goalie in as G. We are interested in the probability of two events R|G and B|G. Here B means the complement of event B, the Bluebirds not scoring. Let's start with using the formula for the probability of a complement to calculate P(B|G).
P(B|G)=1-P(B|G)=1- 0.1= 0.9
Note that the events we are interested in are independent, since scoring by one team does not effect scoring by the other team. Recall the formula for the probability of two independent events.
P(AandB)=P(A)* P(B)
We can use this formula to calculate the desired probability. Let's do it!
The probability that the Redbirds score and the Bluebirds do not score when the coach leaves the goalie in is 0.09, or 9%.
b
Now, we want to find the probability that the Redbirds score and the Bluebirds do not score when the coach takes the goalie out.
Goalie
No Goalie
Redbirds Score
0.1
0.3
Bluebirds Score
0.1
0.6
We are interested in the probability of two events occurring together, R|G and B|G. Here B means the complement of event B, the Bluebirds not scoring. Let's start with using the formula for the probability of a complement to calculate P(B|G).
P(B|G)=1-P(B|G)=1- 0.6= 0.4
Note that the events we are interested in are independent, since scoring by one team does not effect scoring by the other team. Again, we can use the formula for the probability of two independent events.
P(AandB)=P(A)* P(B)
We can use this formula to calculate the desired probability. We will substitute the correct probabilities from the table.
The probability that the Redbirds score and the Bluebirds do not score when the coach takes the goalie out is 0.12, or 12%.
c
Consider the two probabilities. When the coach leaves the goalie in there is 9% chance that the Redbirds will score, and the Bluebirds will not score. When the coach takes the goalie out there is 12% chance that the Redbirds will score, and the Bluebirds will not score. Therefore, taking the goalie out seems a better option, since the probability of scoring is 3% higher.
