Question 1088263
<font color="black" face="times" size="3">V = person is vaccinated
D = person gets disease


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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 <u>given</u> the person is vaccinated". The keyword "given" is important.


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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 <u>given</u> 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)


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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


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Summary so far:


P(D|V) = 0.05
P(D|V') = 0.35
P(V) = 1/3
P(V') = 1/3

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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)


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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) ... <font color=blue>Call this equation (1)</font>


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') ... <font color=green>Call this equation (2)</font>


We'll use <font color=blue>equation (1)</font> and <font color=green>equation (2)</font> in the next section.


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Now we can find P(D).


Using the <a href="https://en.wikipedia.org/wiki/Law_of_total_probability">law of total probability</a>, we can say,


P(D) = P(D and V) + P(D and V')
P(D) = <font color=blue>P(D|V)*P(V)</font> + P(D and V') ... Make a substitution using <font color=blue>equation (1)</font>
P(D) = P(D|V)*P(V) + <font color=green>P(D|V')*P(V')</font> ... Make a substitution using <font color=green>equation (2)</font>
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


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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%


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