Question 1088263: If a person is vaccinated properly, the probability of his getting a certain disease is 0.05. Without a vaccination, the probability of getting the disease is 0.35. Assume that 1/3 of the population is vaccinated. If a person is selected at random from the population, what is the probability of that person's getting the disease?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! V = person is vaccinated
D = person gets disease
----------------------------------------------------
Given Info: "If a person is vaccinated properly, the probability of his getting a certain disease is 0.05"
Translation to symbols: P(D|V) = 0.05
This is conditional probability notation. Asking "what is P(D|V)?" is the same as asking "What is P(D) if we know V is true?"
In English, it means "What is the probability of getting the disease given the person is vaccinated". The keyword "given" is important.
----------------------------------------------------
Given Info: "Without a vaccination, the probability of getting the disease is 0.35"
Translation to symbols: P(D|V') = 0.35
This is the same as "the probability of getting the disease given the person is not vaccinated".
V' is the complement of event V, which means P(V)+P(V') = 1 and P(V') = 1-P(V)
----------------------------------------------------
Given Info: "Assume that 1/3 of the population is vaccinated"
Translation: P(V) = 1/3
This means that P(V') = 1-P(V) = 1-(1/3) = 2/3
----------------------------------------------------
Summary so far:
P(D|V) = 0.05
P(D|V') = 0.35
P(V) = 1/3
P(V') = 1/3
----------------------------------------------------
Given Question: "If a person is selected at random from the population, what is the probability of that person's getting the disease?"
What we want to find: The value of P(D)
----------------------------------------------------
Before we can find P(D), we need to do a bit of side work.
Let's use the definition of conditional probability to rewrite things a bit
P(D|V) = [P(D and V)]/P(V)
P(D|V)*P(V) = P(D and V)
P(D and V) = P(D|V)*P(V) ... Call this equation (1)
P(D|V') = [P(D and V')]/P(V')
P(D|V')*P(V') = P(D and V')
P(D and V') = P(D|V')*P(V') ... Call this equation (2)
We'll use equation (1) and equation (2) in the next section.
----------------------------------------------------
Now we can find P(D).
Using the law of total probability, we can say,
P(D) = P(D and V) + P(D and V')
P(D) = P(D|V)*P(V) + P(D and V') ... Make a substitution using equation (1)
P(D) = P(D|V)*P(V) + P(D|V')*P(V') ... Make a substitution using equation (2)
P(D) = 0.05*(1/3) + 0.35*(2/3) ... plug in the given information (see the "summary so far" section above)
P(D) = 0.25
----------------------------------------------------
----------------------------------------------------
In decimal form, the final answer is 0.25
In fraction form, the final answer is 1/4 (because 1/4 = 0.25)
In percent form, the final answer is 25%
|
|
|
