The probability that a visit to a primary care​ physician's (PCP) office results in neither lab work nor referral to a specialist is 3535​%. Of those coming to a​ PCP's office, 2828​% are referred to specialists and 4343​% require lab work. Determine the probability that a visit to a​ PCP's office results in both lab work and referral to a specialist.  


Let's define the events involved:

- Let \( A \) represent the event that a patient is referred to a specialist.
- Let \( B \) represent the event that a patient requires lab work.
- The probability that a patient gets neither lab work nor a referral to a specialist is given as \( P(\text{not } A \cap \text{not } B) = 35\% \), or 0.35.

We are also given:
- \( P(A) = 28\% \), or 0.28, which is the probability of being referred to a specialist.
- \( P(B) = 43\% \), or 0.43, which is the probability of requiring lab work.

We want to determine the probability that a patient receives both lab work and a referral to a specialist, which is \( P(A \cap B) \).

### Step 1: Use the total probability formula
The total probability of all events must add up to 1. This includes:
- \( P(\text{not } A \cap \text{not } B) \): the probability of neither lab work nor referral (given as 0.35),
- \( P(\text{not } A \cap B) \): the probability of lab work but no referral,
- \( P(A \cap \text{not } B) \): the probability of a referral but no lab work, and
- \( P(A \cap B) \): the probability of both lab work and a referral.

Thus, the total probability is:
\[
P(\text{not } A \cap \text{not } B) + P(\text{not } A \cap B) + P(A \cap \text{not } B) + P(A \cap B) = 1
\]
Substitute \( P(\text{not } A \cap \text{not } B) = 0.35 \), and express the remaining terms in terms of known probabilities:
\[
0.35 + P(\text{not } A \cap B) + P(A \cap \text{not } B) + P(A \cap B) = 1
\]

### Step 2: Calculate probabilities of other events
- The probability of needing lab work is 0.43, so the probability of needing lab work but not being referred is \( P(\text{not } A \cap B) = 0.43 - P(A \cap B) \).
- Similarly, the probability of being referred but not needing lab work is \( P(A \cap \text{not } B) = 0.28 - P(A \cap B) \).

Now, substitute these into the equation:
\[
0.35 + (0.43 - P(A \cap B)) + (0.28 - P(A \cap B)) + P(A \cap B) = 1
\]
Simplify the equation:
\[
0.35 + 0.43 + 0.28 - P(A \cap B) = 1
\]
\[
1.06 - P(A \cap B) = 1
\]
\[
P(A \cap B) = 1.06 - 1 = 0.06
\]

### Conclusion:
The probability that a visit to a primary care physician's office results in both lab work and referral to a specialist is 6%, or \( P(A \cap B) = 0.06 \).

Would you like further details or have any questions?

Here are five related questions to extend this discussion:
1. What is the probability that a patient requires either lab work or a referral (or both)?
2. If 200 patients visit the PCP, how many would you expect to need both lab work and a referral?
3. What is the probability that a patient requires only lab work but not a referral?
4. What is the probability that a patient is referred to a specialist but does not need lab work?
5. How would these probabilities change if the percentage of patients needing lab work increased to 50%?

**Tip:** When calculating probabilities of combined events, use the inclusion-exclusion principle to avoid over-counting intersections.

The probability that a visit to a primary care​ physician's (PCP) office results in neither lab work nor referral to a specialist is 35%. Of those coming to a​ PCP's office, 28% are referred to specialists and 43% require lab work. Determine the probability that a visit to a​ PCP's office results in both lab work and referral to a specialist.

Determine the probability that a visit to a primary care physician results in both lab work and referral to a specialist, given that 28% of patients are referred and 43% require lab work. Learn how to apply the total probability formula and set theory to solve this real-world problem.

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