Recall that if P(B) and P(B|A) are equal, A and B are independent events.
Yes, see solution.
Practice makes perfect
We want to determine if recommending the provider to a friend and living in Santa Monica are independent events. To do so we will use the given two-way table.
At first we will define two events, which will be useful in further calculations.
A — a customer recommends the provider.
B — a customer is located in Santa Monica.We will use the property that if P(B) and P(B|A) are equal, A and B are independent events.
P(B) ? = P(B|A)
To find P(B), which is the probability that the customer is from Santa Monica, we will calculate the marginal relative frequency of the second column of the table.
P(B)=0.27+0.03 ⇔ P(B)= 0.3
To evaluate the right hand side of the equation we will use the formula for the conditional probability.
P(B|A)=P(AandB)/P(A)
We know that a customer is located in Santa Monica and will recommend the provider with a probability of 0.27, which corresponds with P(AandB).
P(AandB)= 0.27
To find P(A), which is the probability that the customer will recommend the provider to a friend, we will calculate the marginal relative frequency of the first row of the table.
P(A)=0.29+0.27+0.32 ⇔ P(A)= 0.88
With this information we can calculate P(B|A).
