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.