Suppose A Class Contains 15 Boys And 30 Girls

Suppose That A Class Contains 15 Boys And 30 Girls And That 10 Studen

Suppose that a class contains 15 boys and 30 girls, and that 10 students are to be selected at random for a special assignment. Find the probability that exactly 3 boys will be selected.

The problem involves calculating the probability of selecting exactly 3 boys out of 10 students chosen randomly from a class comprising 15 boys and 30 girls. To solve this, we can utilize combinatorial probability principles, specifically the hypergeometric distribution, since the selections are made without replacement.

The total number of students in the class is 15 (boys) + 30 (girls) = 45 students. The total number of ways to select 10 students from the entire class is given by the combination:

\[ \binom{45}{10} \]

Next, the favorable outcomes involve selecting exactly 3 boys from the 15 available and 7 girls from the 30 available. The number of ways to do this is:

\[ \binom{15}{3} \times \binom{30}{7} \]

The probability of this event is then:

\[ P(\text{exactly 3 boys}) = \frac{\binom{15}{3} \times \binom{30}{7}}{\binom{45}{10}} \]

Calculating each component using combinations:

- \(\binom{15}{3} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455\)

- \(\binom{30}{7} = \frac{30!}{7! \times 23!}\)

- \(\binom{45}{10} = \frac{45!}{10! \times 35!}\)

Using a calculator or software for factorials to get precise values:

- \(\binom{30}{7} = 2035800\)

- \(\binom{45}{10} = 3,586,261,767\)

Thus, the probability becomes:

\[ P = \frac{455 \times 2,035,800}{3,586,261,767} \approx \frac{927,129,000}{3,586,261,767} \approx 0.2584 \]

Therefore, the probability that exactly 3 boys are selected is approximately 25.84%.

This approach employs the hypergeometric probability because it accurately models the process of choosing without replacement from a finite population. The logic is grounded in the combinatorial reasoning about how many favorable selections exist versus the total possible selections. It ensures an exact probability and aligns with the methodologies commonly used in discrete probability problems involving finite populations.

Understanding why this method is appropriate stems from recognizing that the scenario involves dependent sampling without replacement, which is precisely what the hypergeometric distribution describes. This model facilitates explicit calculations of probabilities for specific configurations within a finite population, providing precise and meaningful insights for decision-making or analysis in real-world contexts such as this classroom scenario.

Paper For Above instruction

The problem of determining the probability that exactly three boys are chosen from a class comprising 15 boys and 30 girls, when 10 students are selected at random, effectively utilises the hypergeometric distribution—a statistical model suitable for sampling without replacement. This approach leverages combinatorial mathematics to calculate the likelihood of a specific composition of the sample, rooted in the principles of probability theory and combinatorics.

The fundamental reasoning hinges on understanding the total number of possible ways to select 10 students from a class of 45 students, which is represented by the combination \(\binom{45}{10}\). This total serves as the denominator in the probability calculation, representing all possible groups of 10 students regardless of composition. The numerator accounts for the favorable outcomes where exactly 3 boys and 7 girls are selected, computed by multiplying the number of ways to choose 3 boys from 15 (\(\binom{15}{3}\)) by the number of ways to select 7 girls from 30 (\(\binom{30}{7}\)).

The calculation of these combinations involves factorial functions, which count the number of ways to choose subsets without regard to order. The value \(\binom{15}{3}\) is straightforward, calculated as 455, reflecting the number of groups of 3 boys possible from the 15. The term \(\binom{30}{7}\) is 2,035,800, representing the possible groups of 7 girls. The total possible combinations for selecting any 10 students from the class is approximately 3.59 billion.

The probability is obtained by dividing the favorable outcomes by the total outcomes, yielding roughly 0.2584 or 25.84%. This precise computation reveals the likelihood of randomly selecting exactly 3 boys during such an experiment, grounded in the principles of hypergeometric probability. It exemplifies how combinatorial reasoning and understanding of sampling processes underpin solutions in finite discrete probability problems, especially in educational or practical scenarios where fixed populations and specific sample conditions are involved.

The choice of the hypergeometric distribution over other probability models such as the binomial is essential because it models dependent sampling scenarios where the population size is finite and the sample is drawn without replacement. Unlike the binomial distribution, which assumes independence with replacement, the hypergeometric directly accounts for the changing probabilities as items are selected, making it more accurate for this classroom example.

In conclusion, this method demonstrates the power of combinatorics and probability theory in solving real-world problems involving finite populations. It emphasizes the importance of understanding the context and the sampling process to select the appropriate model, thereby ensuring precise and meaningful probability estimates.

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