a If two events are independent, then the probability of both events occurring is the product of the probabilities of the events.
B
b The conditional probability of an event B occurring, given that event A has occurred previously, is the quotient of the probability of both events occurring and the probability of event A.
A
a 5 %, see solution.
B
b 75 %
Practice makes perfect
a We know that in a small town in England, 75 % of homes have large backyards and 80 % of homes have garages. Let's assume that these characteristics are independent. We want to find the probability that a home in this town has neither a garage nor a large backyard.
P(neither a garage nor a large backyard)= ?
Let's use a generic area model to find the answer. We will start by finding the probabilities of not having a garage and of not having a large backyard. Since a house can either have a garage or not have one, the probability of not having a garage will be 100 % minus the probability of having a garage. The same applies to having a large backyard.
P(no garage)= & 100 %- 75 %= 25 %
P(no large backyard)= & 100 %- 80 %= 20 %
Now let's create the generic area model for these two independent events. Remember, the two variables in this situation are not having a garage and not having a large backyard!
Next, let's fill in our model. We do this by multiplying the probability in the row by the probability in the column. We follow this process because the probability of two independent events occurring is the product of the probabilities of each event.
From our model, we can see that the probability that a home has neither a garage nor a large backyard is 5 %.
b We want to find the conditional probability that a house has a large backyard if it has a garage. Let's recall that P(B|A) — the conditional probability of an event B occurring given that event A has already occurred — is the quotient of the probability of both events occurring and the probability of event A.
P(B|A)=P(AandB)/P(A)
Here, event A is a house having a garage and event B is the same house having a large backyard. Let's use the area model we created in Part A to find P(A) and P(AandB).
Now let's substitute the probabilities into the formula for the conditional probability. We will rewrite the percentages into decimals.
P(B|A)=P(AandB)/P(A)=0.60/0.80
Let's evaluate the quotient!
