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 a patient visits a physical therapist. Calculate the probability that a randomly chosen member of this group visits a physical therapist

To solve this problem, we'll use the principles of probability and the relationships between different events. Let's define the following:

- \( P(P) \): The probability that a patient visits a physical therapist.
- \( P(C) \): The probability that a patient visits a chiropractor.
- \( P(P \cap C) \): The probability that a patient visits both a physical therapist and a chiropractor. This is given as 22%, or \( P(P \cap C) = 0.22 \).
- \( P(\text{neither}) \): The probability that a patient visits neither a physical therapist nor a chiropractor. This is given as 12%, or \( P(\text{neither}) = 0.12 \).
- \( P(P \cup C) \): The probability that a patient visits either a physical therapist, a chiropractor, or both.

Using the fact that the probability of the union of two events \( P(P \cup C) \) is related to the probability of each event and their intersection, we have:

\[
P(P \cup C) = P(P) + P(C) - P(P \cap C)
\]

Since \( P(\text{neither}) = 1 - P(P \cup C) \), we can write:

\[
P(\text{neither}) = 1 - (P(P) + P(C) - P(P \cap C))
\]

Substituting the given values:

\[
0.12 = 1 - (P(P) + P(C) - 0.22)
\]

This simplifies to:

\[
P(P) + P(C) = 1 - 0.12 + 0.22 = 1.10
\]

We are also given that the probability of visiting a chiropractor exceeds the probability of visiting a physical therapist by 0.14:

\[
P(C) = P(P) + 0.14
\]

Substituting this into the equation \( P(P) + P(C) = 1.10 \):

\[
P(P) + (P(P) + 0.14) = 1.10
\]

This simplifies to:

\[
2P(P) + 0.14 = 1.10
\]

Subtracting 0.14 from both sides:

\[
2P(P) = 0.96
\]

Dividing by 2:

\[
P(P) = 0.48
\]

So, the probability that a randomly chosen patient visits a physical therapist is \( \boxed{0.48} \).

Would you like further details or have any questions?

Here are 5 related questions you might consider:
1. What is the probability that a patient visits only a chiropractor?
2. How can you calculate the probability that a patient visits only a physical therapist?
3. What is the total probability that a patient visits at least one of the two specialists?
4. How would the solution change if the probabilities of visiting a physical therapist and a chiropractor were equal?
5. How would the result change if the probability of visiting neither specialist was different?

**Tip:** In probability problems involving two events, always consider both their union and intersection to fully understand the relationships between them.

Explore how to calculate the probability that a patient visits a physical therapist in a group recovering from shoulder injuries. This problem involves understanding probabilities, set theory, and intersections of events. Detailed step-by-step solution provided.

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