SOLUTION: The probability that a person selected at random from a population will exhibit the classic symptom of a certain disease is 0.20 and the probability that a person selected at rando

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Question 1084188: The probability that a person selected at random from a population will exhibit the classic symptom of a certain disease is 0.20 and the probability that a person selected at random has the disease is 0.23 the probability that a person who has the symptoms also has the disease is 0.18 A person is selected at random from a population does not have the symptoms.what is the probability that the person has the disease??
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

Define the following events:
S = person exhibits symptoms (they may or may not actually have the disease)
D = person actually has the disease (they may or may not show symptoms)

Those are simple events which can be combined to form this compound event
S and D = person has symptoms AND person actually has disease

We're given the following probabilities
P(S) = 0.2
P(D) = 0.23
P(S and D) = P(D and S) = 0.18

We want to find the conditional probability P(D|S') which is another way of asking "what is P(D) given that S' has occured?".
A translation: "What is the probability of getting a person with the disease given they do not show symptoms?"

Note: The notation S' is the complementary event of event S. If S doesn't happen, then S' does happen and vice versa. They are opposite one another. Since
S = person exhibits symptoms
this means
S' = person does not show symptoms
Your book may use the notation instead of S' to denote a complementary event. As you can probably guess, the 'c' in the exponent means "complementary".

The conditional probability formula we'll use is
P(D|S') = P(D and S')/P(S')

so we'll first need to find P(S') and P(D and S')

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Let's find the probability of not showing symptoms P(S')
P(S) + P(S') = 1
P(S') = 1 - P(S)
P(S') = 1 - 0.2
P(S') = 0.8

Use the Law of Total Probability to find P(D and S')
P(D) = P(D and S) + P(D and S')
0.23 = 0.18 + P(D and S')
P(D and S') = 0.05

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Now we can compute the answer
P(D|S') = P(D and S')/P(S')
P(D|S') = 0.05/0.8
P(D|S') = 0.0625

The probability is 0.0625

Notes:
0.0625 = 625/10000 = 1/16 is the fraction form of the answer
0.0625*(100%) = 6.25% is the percent form of the answer
the percent, decimal and fraction forms all represent the same answer, just in different ways