a How many axes do you need when making an area model that demonstrates three events?
B
b Each coin flip can have two outcomes.
C
c Mark the paths through the diagram that result in the desired outcome.
D
d How many paths result in each of the outcomes?
A
a A tree diagram is better. See solution.
B
bDiagram:
Possible Outcomes: 8
C
c P(3 heads)= 18
P(1 head and 2 tails)= 38
P(at least 1tail)= 78
P(exactly 2 tails)= 38
D
d Equally likely. See solution.
Practice makes perfect
a An area model is only good for outcomes with two events. When there are three events the area model breaks down you need a third axis to demonstrate it. However, since the model is rather simple we do not need a systematic list, which is better for more complex situations. Therefore, we should use a tree diagram.
b A coin flip has two outcomes: heads or tails. Therefore, we will have a two-way split for each level of our tree diagram.
We can see that there are 8 possible outcomes.
c We are now asked to determine the probability of some events occurring. It is equally likely to flip heads as it is to flip tails. Therefore, heads and tails both have a 12 probability of happening.
Three Heads
Let's highlight the path where all three coins show heads.
The probability of getting three heads is the product of the probabilities along the path of the tree diagram.
P(3 heads): 1/2*1/2*1/2=1/8
One Head and Two Tails
Let's highlight the paths where we get one head and two tails.
We have three paths that give the desired outcome. Like previously, we can calculate the probability of each outcome by multiplying the probabilities along each path of the tree diagram.
P(tail, tail, head): 1/2*1/2*1/2=1/8 [0.8em]
P(tail, head, tail): 1/2*1/2*1/2=1/8 [0.8em]
P(head, tail, tail): 1/2*1/2*1/2=1/8
To calculate the probability to get any of these events, we add their probabilities.
P(1 head and 2 tails): 1/8+1/8+1/8=3/8
At Least One Tail
Note that at least one tail is the complement to the probability of getting three heads. Therefore, we can calculate this by subtracting P(3 heads) from 1.
Exactly two tails is the same outcome as getting two tails and a head. We have already determined this probability as 38.
d Let's highlight the paths that result in the different requested outcomes.
We will also show outcomes that result in at least 2 tails.
The number of paths through the tree diagram resulting in at least 2 heads and at least 2 tails is the same. Since all paths show a probability of 12, the probability of these events must be the same.
