b Calculate the conditional probability of being located in Glens Falls for a not satisfied patient.
C
c To determine whether events A and B are independent, we have to check if the expressions P(AandB) and P(A)* P(B) are equal.
A
a About 0.789
B
b 0.168
C
c The events are independent.
Practice makes perfect
a We want to find the probability that a randomly selected patient from the hospital Saratoga was satisfied with the physician's communication. To do so, we will use the given two-way table with the results of the survey.
At first we will define two events, which will be useful in further calculations.
A — a patient was located in Saratoga.
B — a patient was satisfied.The probability that an event will occur given that another event has already occurred is called a conditional probability. Therefore, we are interested in P(Yes|Saratoga), which can be denoted as P(B|A). Let's recall the formula for the conditional probability.
P(B|A)=P(AandB)/P(A)
We know that a patient was located in Saratoga and was satisfied with a probability of 0.288, which corresponds with P(AandB).
P(AandB)= 0.288
To find P(A), which is the probability that the patient was located in Saratoga, we will calculate the marginal relative frequency of the second column.
P(A)=0.288+0.077
⇕
P(A)= 0.365
With this information, we can calculate P(B|A).
The probability that a randomly selected patient located in Saratoga was satisfied with the physician's communication is about 0.789.
b We want to find the probability that a randomly selected patient that was not satisfied with the physician's communication was located in the hospital in Glens Falls. To do so we will use the given two-way table with the results of the survey.
First we will define two events, which will be useful in further calculations.
A — a patient was not satisfied.
B — a patient was located in Glen Falls.Therefore, we are interested in calculating the conditional probability P(Glen Falls|No), which can be denoted as P(B|A). Let's recall the formula for the conditional probability.
P(B|A)=P(AandB)/P(A)
We know that a patient was not satisfied and was located in Glen Falls with a probability of 0.042, which corresponds with P(AandB).
P(AandB)= 0.042
To find P(A), which is the probability that the patient was not satisfied, we will calculate the marginal relative frequency of the second row.
P(A)=0.042+0.077+0.131
⇕
P(A)= 0.25
With this information, we can calculate P(B|A).
The probability that a randomly selected patient that was not satisfied with the physician's communication was located in Glen Falls is 0.168.
c We want to determine whether being satisfied and located in Saratoga are independent events. To do so, we will use the given two-way table with the results of the survey.
First we will define two events, which will be useful in further calculations.
A — a patient was located in Saratoga.
B — a patient was satisfied.We will use the property that if P(A)* P(B) and P(AandB) are equal, A and B are independent events.
P(A)* P(B) ? = P(AandB)
To find P(A), which is the probability that the patient was located in Saratoga, we will calculate the marginal relative frequency of the second column of the table.
P(A)=0.288+0.077
⇕
P(A)= 0.365
To find P(B), which is the probability that the patient was satisfied with the communication of the physician, we will calculate the marginal relative frequency of the first row of the table.
P(B)=0.123+0.288+0.338
⇕
P(B)= 0.749
With this information we can calculate P(A)* P(B).
Finally, we know that a patient was satisfied and was located in Saratoga with the probability of 0.288, which corresponds with P(AandB). Let's round this value to the nearest tenth.
P(AandB)= 0.288
⇕
P(AandB) ≈ 0.3
We obtained that both P(A)* P(B) and P(AandB) are approximately 0.3, so being satisfied with the communication of the physician and being located in Saratoga are independent events.
