Conditional Probability?

Conditional probability question.





A new drug was administered to 100 people. Follow-up interviews were conducted in which patients were asked whether or not they experienced various side effects. 7 of the patients reported experiencing a rise in blood pressure, 6 of the patients reported experiencing a loss of sleep, and 3 reported experiencing both of these side effects. According to these results, what is the probability that a person taking this drug





(a) will experience a rise in blood pressure, given that the person experiences a loss of sleep?





(b) will experience loss of sleep, given that the person experiences a rise in blood pressure?





(c) will experience both of these symptoms, given that the person experiences at least one of these symptoms?

Conditional Probability?
The general answer for conditional probabilities is:





P(A∩B) = P(A|B)P(B)





a) P(loss of sleep) = 0.09 (a total of 9 patients experimented a loss of sleep: 6 had that symptom only, and 3 had both)


P(loss of sleep ∩ high blood pressure) = 0.03 (3 patients had both a loss of sleep and a rise in blood pressure)


P(rise in blood pressure| loss of sleep) = 0.03/0.09 = 1/3 = 33.33%





b) P(high blood pressure) = 0.1 (a total of 10 patients experimented a rise in blood pressure: 7 had that symptom only, and 3 had both)


P(loss of sleep ∩ high blood pressure) = 0.03 (3 patients had both a loss of sleep and a rise in blood pressure)


P(loss of sleep | rise in blood pressure) = 0.03/0.1 = 0.3 = 30%





c) P(at least one symptom) = 0.07 + 0.06 + 0.03 = 0.16 (16 patients reported at least one symptom)


P(at least one symptom ∩ both symptoms) = P(both symptoms) = 0.03 The intersection of the two sets is the set of people with both symptoms.


P(both symptoms | at least one symptom) = 0.03/0.16 = 3/16 = 0.1875 or 18.75%