Clinic records show that 24% of all patients coming in have strep throat and 53%

Question

Clinic records show that 24% of all patients coming in have strep throat and 53% of all patients have an allergy. Included in those numbers are the 10% of patients that have both strep throat and an allergy.
( a). What is the probability that a patient has at least one of these two problems?    (b). What proportion of those with strep throat also have an allergy? (c). What proportion of those with an allergy also have strep? (d.) Are the events “has strep throat” and “has an allergy” mutually exclusive? Why or why not?    (e). Are the events “has strep throat” and “has an allergy” independent? Justify your answer with probability.

Explanation / Answer

S: Strep throat and P(S)=0.24

A: allergy and P(A)=0.53

P(S and A)=0.1

a) P( S or A)=P(A)+P(S)-P(A and S) =0.24+0.53-0.1=0.67

b) P(A|S)=P(A and S)/P(S)=0.1/0.24=0.4167

c) P(S|A)=P(A and S)/P(A)=0.1/0.53=0.188

d) For events to be mutually exlusive, P(S or A) should be P(A)+P(S) but it is not the case here so NO

e) For events to be independent, P(A and S)=P(A)*P(S) should be satisfied which is not the case here, so NO

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