A group of 5 boys and 10 girls is lined up in random order — that is, each
of the 15! permutations is assumed to be equally likely.
a. What is the probability that the person in the 4th position is a boy?

To solve this problem, let's define the situation more clearly:

- There are 5 boys and 10 girls, making a total of 15 people.
- We are interested in finding the probability that the person in the 4th position of the lineup is a boy.

### Solution:

1. **Determine the total number of possible lineups**:
   
   Since the lineup is random, each of the 15 people can occupy any of the 15 positions. Therefore, the total number of permutations of these 15 people is:

   \[
   15! = 15 \times 14 \times \ldots \times 2 \times 1
   \]

2. **Probability of the 4th position being a boy**:

   Since each person has an equal probability of being in any position, the probability of any specific position being occupied by a boy is the ratio of the number of boys to the total number of people.

   The 4th position can be filled by any of the 5 boys. So, the probability that a boy occupies this position is:

   \[
   \text{Probability} = \frac{\text{Number of boys}}{\text{Total number of people}} = \frac{5}{15}
   \]

   Simplifying this fraction:

   \[
   \frac{5}{15} = \frac{1}{3}
   \]

### Final Answer:

The probability that the person in the 4th position is a boy is:

\[
\boxed{\frac{1}{3}}
\]

Would you like further details or have any other questions?

### Related Questions:

1. What is the probability that the first position is occupied by a girl?
2. What is the probability that the 4th position is occupied by a girl?
3. If the person in the 4th position is a boy, what is the probability that the next position (5th) is also a boy?
4. What is the probability that boys occupy all odd-numbered positions?
5. How would the probability change if there were 6 boys and 9 girls instead?

**Tip:** Probabilities can often be simplified by looking at ratios, especially when each outcome is equally likely.

A group of 5 boys and 10 girls is lined up in random order — that is, each of the 15! permutations is assumed to be equally likely. What is the probability that the person in the 4th position is a boy?

Learn how to calculate the probability that a boy is in the 4th position in a lineup of 5 boys and 10 girls, with a detailed step-by-step solution. This problem involves key probability concepts and permutations. Suitable for students in Grades 9-12.

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