Suppose that 7 out of the 19 doctors in a small hospital are General Practitioners, 8 out of the 19 are under the age of 45, and 2 are both General Practitioners and under the age of 45. What is the probability that you are randomly assigned a General Practitioner or a doctor under the age of 45? Enter a fraction or round your answer to 4 decimal places, if necessary.

To solve this problem, we can use the principle of inclusion and exclusion to find the probability that a randomly assigned doctor is either a General Practitioner (GP) or under the age of 45.

### Step 1: Define the events
- Let \( A \) be the event that the doctor is a General Practitioner (GP).
- Let \( B \) be the event that the doctor is under the age of 45.

We are given:
- The total number of doctors is \( 19 \).
- The number of General Practitioners (event \( A \)) is \( |A| = 7 \).
- The number of doctors under the age of 45 (event \( B \)) is \( |B| = 8 \).
- The number of doctors who are both General Practitioners and under the age of 45 (i.e., \( A \cap B \)) is \( |A \cap B| = 2 \).

### Step 2: Apply the inclusion-exclusion principle
To find the probability of either being a General Practitioner or under 45 (i.e., \( P(A \cup B) \)), we use the formula:

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

The individual probabilities are:
- \( P(A) = \frac{|A|}{19} = \frac{7}{19} \)
- \( P(B) = \frac{|B|}{19} = \frac{8}{19} \)
- \( P(A \cap B) = \frac{|A \cap B|}{19} = \frac{2}{19} \)

### Step 3: Calculate the total probability
Now, substitute these values into the inclusion-exclusion formula:

\[
P(A \cup B) = \frac{7}{19} + \frac{8}{19} - \frac{2}{19} = \frac{7 + 8 - 2}{19} = \frac{13}{19}
\]

Thus, the probability that you are randomly assigned a General Practitioner or a doctor under the age of 45 is:

\[
P(A \cup B) = \frac{13}{19}
\]

### Final Answer:
The probability is \( \frac{13}{19} \), which is approximately \( 0.6842 \) when rounded to 4 decimal places.

Would you like further details on this solution or have any questions?

Here are 5 related questions you can explore:
1. What is the probability that a randomly assigned doctor is **neither** a General Practitioner nor under the age of 45?
2. If we randomly select two doctors, what is the probability that both are General Practitioners?
3. What is the probability that a randomly selected doctor is **only** a General Practitioner (not under 45)?
4. If we know that a doctor is under the age of 45, what is the probability that they are also a General Practitioner?
5. What is the probability that a doctor is both a General Practitioner and under the age of 45?

**Tip:** When dealing with probabilities involving "or," inclusion-exclusion helps account for any overlap between events.

Suppose that 7 out of the 19 doctors in a small hospital are General Practitioners, 8 out of the 19 are under the age of 45, and 2 are both General Practitioners and under the age of 45. What is the probability that you are randomly assigned a General Practitioner or a doctor under the age of 45? Enter a fraction or round your answer to 4 decimal places, if necessary.

Learn how to calculate the probability of being assigned a General Practitioner or a doctor under 45 in a small hospital. Explore key probability concepts such as inclusion-exclusion principle and solve the problem using step-by-step methods. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation