Alyssa Wants To Know What Students Think About Replacing The
Alyssa Wants To Know What Students Think About Replacing The Candy In
Alyssa wants to know what students think about replacing the candy in two vending machines with more healthy snacks. There are 300 sixth graders, 300 seventh graders, and 200 eighth graders. Half of the students are girls. Alyssa obtains a list of student names, grouped by grade, with the girls listed first in each grade. Alyssa randomly chooses 3 different names from the list of 800 students. What are the chances the first choice is a girl? Second choice is a girl? Third choice is a girl?
Paper For Above instruction
The problem at hand involves understanding the probability of selecting girls from a list of students with specific given conditions. To analyze such probabilities accurately, it is crucial to understand the composition of the student body and the process of random selection.
Student Population Composition
There are three grade levels: sixth, seventh, and eighth grades, with 300, 300, and 200 students respectively, totaling 800 students (300 + 300 + 200 = 800). Since half of the students are girls, the total number of girls in the entire school is 400 (half of 800).
The list of student names is arranged such that girls are listed first within each grade, then boys. This ordered arrangement affects the probability calculations only if the selection process is influenced by the order. However, in random selection, each student has an equal chance of being chosen regardless of their position in the list.
Probability of the First Selection Being a Girl
Since the student list includes 400 girls out of 800 students, if Alyssa randomly picks a student with equal probability, the chance the first selected student is a girl is simply the ratio of girls to total students:
\[ P(\text{first is girl}) = \frac{400}{800} = \frac{1}{2} \]
This probability assumes independent random selection without replacement, and equally likely choice among all students.
Probability of the Second Selection Being a Girl
Given that the first student has been selected and was, say, a girl, then there are now 399 girls remaining out of 799 remaining students in total. The probability that the second choice is a girl, therefore, depends on whether the first was a girl or a boy:
- If the first was a girl (probability ½), then the probability the second is a girl is:
\[ \frac{399}{799} \]
- If the first was not a girl (also probability ½), then the number of girls remains at 400, but total students reduce to 799, so the probability the second is a girl in this case is:
\[ \frac{400}{799} \]
Since the initial probability of the first being a girl is ½, the overall probability the second is a girl involves considering both the scenarios:
\[ P(\text{second is girl}) = P(\text{first was girl}) \times \frac{399}{799} + P(\text{first was boy}) \times \frac{400}{799} \]
\[ P(\text{second is girl}) = 0.5 \times \frac{399}{799} + 0.5 \times \frac{400}{799} \]
\[ P(\text{second is girl}) = \frac{1}{2} \times \frac{399 + 400}{799} = \frac{1}{2} \times \frac{799}{799} = \frac{1}{2} \]
Thus, the probability the second chosen student is a girl remains ½.
Probability of the Third Selection Being a Girl
Similarly, after two selections, the probability that the third choice is a girl can be computed using the same logic and symmetry. The probability that the third choice is a girl turns out to be:
\[ P(\text{third is girl}) = \frac{1}{2} \]
assuming the process involves random selection without replacement, and the initial proportions are balanced at 50%.
Conclusion
Since the total number of girls and boys is equal, and each selection is random without replacement, the probability that each of the first, second, and third choices is a girl remains consistent at ½. This is an application of the symmetry inherent in sampling without replacement from a balanced population.
Implications of the Problem
These calculations underscore the importance of understanding basic probability principles when dealing with random selections from equally balanced populations. The fact that the list is sorted with girls listed first in each grade does not influence the probability unless selection biases are introduced, which is not indicated here.
References
- Ross, S. M. (2014). Introduction to Probability Models (11th ed.). Academic Press.
- Frank, R. (2009). Statistics and Probability for Engineering and the Sciences. Pearson.
- Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
- Moore, D. S., & McCabe, G. P. (2017). Introduction to the Practice of Statistics. W.H. Freeman.
- Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury Press.