Among a large group of patients recovering from shoulder injuries, it is found t

Question

Among a large group of patients recovering from shoulder injuries, it is found that 19% visit both a physical therapist and a chiropractor, whereas 12% visit neither of these. The probability that a patient visits a chiropractor exceeds by 0.17 the probability that a patient visits a physical therapist. Determine the probability that a randomly chosen member of this group visits a physical therapist The probability that a member of this group visits a physical therapist is Type an integer or a decimal.)

Explanation / Answer

Let P(A) denote the probability that a patient visits a pysical therapist and P(B) denote the probability that the patient visits a chiropractor.

We have P(A and B) = 0.19 P(A' and B') = 0.12

Using De-Morgan's law

P(A or B) = 1 - P(A' and B') = 1 - 0.12 = 0.88

We have P(A) + P(B) - P(A and B) = P(A or B)

=> P(A) + P(B) - 0.19 = 0.88

=> P(A) + P(B) = 0.88 + 0.19

=> P(A) + P(B) = 1.07

Also given is P(B) - P(A) = 0.17

Adding

=> 2 P(B) = 1.07 + 0.17

=> 2 P(B) = 1.24

=> P(B) = 1.24 / 2 = 0.62

Substituting in the earlier equation

=> P(A) + 0.62 = 1.07

=> P(A) = 1.07 - 0.62 = 0.45.

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