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Among a large group of patients recovering from shoulder injuries, it is found that 22% Visit both a physical therapist and a chiropractor, whereas 12% visit neither of these. The probability that a patient visits a chiropractor exceeds by 0.14 the probability that d le to 010.0 validedo a patient visits a physical therapist. oblodiog bemion-l dans h an n itidadong Determine the probability that a randomly chosen member of this group visits a physical therapist.

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Answer:

0.48

Step-by-step explanation:

First of all. let's recall probability formula needed for solving the problem.

We know that, If the events A and B are not mutually exclusive, the probability is:  

probability of event (A or B) = probability of event A + probability of event B – p(A and B).

This formula will be used in calculations of this problem.

Let's consider that event A is patient visiting physical therapist; event B is patient visiting chiropractor.

P (A∩B) = 0.22 (visiting both physical therapist and chiropractor) ⇔ p(A and B)

P(B) = P(A) + 0.14 [ Probability patient visiting chiropractor is 0.14 more than probability visiting physical therapist]

Patients visiting none of these is 12%

Those who visit either therapist or chiropractor are amount to 1 - 0.12 = 0.88

Now we should use the formula mentioned in the beginning of the text:

0.88 = P(A) + P(B) - 0.22 ⇒ P(A) + P(B) = 1.1

If we replace P(B) by P(A) + 0.14 ⇒ 2P(A) + 0.14 = 1.1 ⇒ 2P(A) = 1.1 - 0.14 = 0.96

So, P(A) = 0.96/2 = 0.48 [Probability of patients visiting therapist]

Probability is the chance of an event occurring. The probability that a patient visits a therapist is 0.48.

What is probability?

The probability is the chance of an event occurring.

We know that the probability of any one of the events occurring can be given by the following formula, therefore,

Probability(A or B) = Probability(A) + Probability(B) – Probability(A and B).

[tex]P(A\cup B) = P(A) + P(B) -P(A\cap B)[/tex]

This is equation 1.

As the probability of a person visiting both a physical therapist and chiropractor is given, therefore, Probability(A and B) is 22%.

Also, it is given that the probability that a patient visits a chiropractor exceeds by 0.14 the probability that the patient visits a physical therapist. Therefore,

[tex]P(B) = P(A) + 0.14[/tex]

Since it is mentioned that the chances of Patients visiting none of these are 12%. Thus, the chances that a patient visits either therapist or chiropractor can be given as 1 - 0.12 = 0.88 [P(A∪B)].

Substitute the values, in equation 1,

[tex]P(A\cup B) = P(A) + P(B) -P(A\cap B)\\\\0.88 = P(A) + P(B) - 0.22\\\\ P(A) + P(B) = 1.1[/tex]

But, P(B)=P(A)+0.14, therefore,

[tex]P(A)+P(B) = 1.1\\\\2P(A) + 0.14 = 1.1 \\\\2P(A) = 1.1 - 0.14 \\\\2P(A)= 0.96\\\\P(A)= 0.48[/tex]

Hence, the probability that a patient visits a therapist is 0.48.

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