Age group

Number in that age group

30 – 39

10

40 – 49

20

50 – 59

30

60 – 69

15

If we selected 14 teachers, what is the probability that 2 are in their 30s, 3 are in their 40s and 9 are in their 50s? Type your answer to 4 decimal places.

To calculate the probability of selecting 14 teachers with specific numbers from different age groups, we can use the **multinomial probability formula**. The formula is as follows:

\[
P = \frac{\binom{n_{30}}{2} \binom{n_{40}}{3} \binom{n_{50}}{9} \binom{n_{60}}{0}}{\binom{n_{total}}{14}}
\]

Where:
- \( n_{30} = 10 \) is the number of teachers in the 30-39 age group,
- \( n_{40} = 20 \) is the number of teachers in the 40-49 age group,
- \( n_{50} = 30 \) is the number of teachers in the 50-59 age group,
- \( n_{60} = 15 \) is the number of teachers in the 60-69 age group,
- \( n_{total} = 10 + 20 + 30 + 15 = 75 \) is the total number of teachers,
- We are selecting a total of 14 teachers: 2 from the 30s, 3 from the 40s, and 9 from the 50s.

Now, we will calculate this probability step by step.

Let me compute this.The probability of selecting 14 teachers such that 2 are in their 30s, 3 are in their 40s, and 9 are in their 50s is \( 0.0013 \), or 0.13%.

Do you have any further questions or would you like more details on how this was calculated?

Here are 5 related questions you could explore:
1. How would the probability change if you selected 10 teachers instead of 14?
2. What is the probability of selecting 1 teacher from each age group in a group of 4?
3. How does the binomial coefficient help in probability calculations?
4. If the number of teachers aged 50-59 increased, how would that affect the probability?
5. What is the probability that all selected teachers are aged 40-49?

**Tip:** The multinomial coefficient helps when you're selecting items from multiple groups simultaneously, considering different proportions for each group.

If we selected 14 teachers, what is the probability that 2 are in their 30s, 3 are in their 40s, and 9 are in their 50s?

Calculate the probability of selecting 14 teachers, with 2 in their 30s, 3 in their 40s, and 9 in their 50s, using the multinomial probability formula. This problem explores key concepts in probability and combinatorics, suitable for advanced high school or college students.

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