We are told that 31 % of Americans recycle. This means that the probability that a randomly chosen American recycles is also 31 %, or 0.31.
P(American recycles)=0.31
Additionally, we are told that two Americans are chosen at random from a group of 50. We are asked about the probability that at most one of them recycles.
P(At most one American recycles)=?
First, let's look at all possible events.
Possible Events
1.
Only the first American recycles
2.
Only the second American recycles
3.
Neither of the Americans recycle
4.
The first and the second Americans recycle
Next, notice that first three events can be combined into one event — at most one American recycles.
Possible Events
1.
At most one American recycles
2.
3.
4.
The first and second Americans recycle
We found that the events at most one American recycles and the first and the second Americans recycle split all possible events into two and therefore are complementary!
Event
Complement
at most one American recycles
the first and second Americans recycle
To calculate the probability that at most one American recycles, it will be easier for us to first calculate the probability of its complement, which is the event where both of the selected Americans recycle.
P(the first and second Americans recycle)=?
These events are independent, and therefore we can calculate this probability using the formula for the probability of independent events.
Probability of Independent Events
If two events A and B are independent, P(AandB) = P(A) * P(B).
Let's do it!
P(the first and second Americans recycle)=
P(first Am. recycles) * P(second Am. recycles)
Since both the first and second Americans were randomly chosen, both of the probabilities in the product are equal to 31 %, or 0.31. Let's substitute!
P(the first and second Americans recycle)=
0.31* 0.31=0.0961
The probability that both of the Americans recycle is 0.0961. Now, we can use the formula for the probability of a complement to calculate the probability that at most one American recycles.
Probability of a complement
For an event A, P(notA)=1-P(A)
Let's call A the event where both of the Americans recycle. Earlier we found that its complement (not A) is the event where at most one of them recycles.
Event
Description
Probability
A
the first and second Americans recycle
0.0961
Not A
at most one American recycles
P(not A)=1-P(A)
Finally, we can calculate the probability that at least one of the Americans recycles, P(not A).
