A realty company has 100 homes listed for sale. Some of these homes have fireplaces, some have garages, and some have neither. A breakdown is provided in the two-way classification table below.
Fireplace No Fireplace
Garage 50 30
No Garage 10 10
what is the probability that the home selected has a garage, given that it has a fireplace?
b. Are the events Fireplace and Garage independent? Show your proof.
2 marks c. Are the events F and G mutually exclusive? Why?
P(G and F)= P(G) P(F/G) =(80/100)(50/80)=0.8x0.625=0.5. a)They are not independent because the probabilities are not equal. P(F)= 60/100=.6, P(F/G)=60/80=0.75
b)Events F and G can occur together because they are not mutually exclusive.
To find the probability that the home selected has a garage, given that it has a fireplace, we need to use conditional probability.
Conditional probability is calculated using the formula:
P(Garage|Fireplace) = P(Garage and Fireplace) / P(Fireplace)
From the given table, we can see that the number of homes with both a fireplace and a garage is 50. The total number of homes with a fireplace (regardless of whether they have a garage or not) is 80 (50 + 30).
Therefore, P(Garage|Fireplace) = 50 / 80 = 0.625
So, the probability that the home selected has a garage, given that it has a fireplace, is 0.625 or 62.5%.
b. To check if the events Fireplace and Garage are independent, we need to compare the joint probability of the events with their individual probabilities.
P(Fireplace) = (Number of homes with a fireplace) / (Total number of homes)
= 80 / 100
= 0.8
P(Garage) = (Number of homes with a garage) / (Total number of homes)
= (50 + 30 + 10 + 10) / 100
= 0.1 + 0.3 + 0.1 + 0.1
= 0.6
The joint probability of Fireplace and Garage is 50 / 100 = 0.5
If the events Fireplace and Garage are independent, then:
P(Fireplace and Garage) = P(Fireplace) * P(Garage)
However, in this case, 0.5 is not equal to 0.8 * 0.6 = 0.48. Therefore, the events Fireplace and Garage are not independent.
c. The events Fireplace and Garage are not mutually exclusive. This is because there are 50 homes that have both a fireplace and a garage. For two events to be mutually exclusive, they should not be able to occur at the same time. In this case, we have homes that satisfy both conditions, so the events are not mutually exclusive.