There are n students at a certain school, of whom X ∼Bin(n, p) are Statistics majors. A simple r

There are n students at a certain school, of whom X ∼Bin(n,...

There are $n$ students at a certain school, of whom $X \sim \operatorname{Bin}(n, p)$ are Statistics majors. A simple random sample of size $m$ is drawn ("simple random sample" means sampling without replacement, with all subsets of the given size equally likely). (a) Find the PMF of the number of Statistics majors in the sample, using the law of total probability (don't forget to say what the support is). You can leave your answer as a sum (though with some algebra it can be simplified, by writing the binomial coefficients in terms of factorials and using the binomial theorem). (b) Give a story proof derivation of the distribution of the number of Statistics majors in the sample; simplify fully. Hint: Does it matter whether the students declare their majors before or after the random sample is drawn?

There are $n$ students at a certain school, of whom $X \sim \operatorname{Bin}(n, p)$ are Statistics majors. A simple random sample of size $m$ is drawn ("simple random sample" means sampling without replacement, with all subsets of the given size equally likely).
(a) Find the PMF of the number of Statistics majors in the sample, using the law of total probability (don't forget to say what the support is). You can leave your answer as a sum (though with some algebra it can be simplified, by writing the binomial coefficients in terms of factorials and using the binomial theorem).
(b) Give a story proof derivation of the distribution of the number of Statistics majors in the sample; simplify fully.

Hint: Does it matter whether the students declare their majors before or after the random sample is drawn?

Key Concepts

Binomial Distribution

The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In the context of this problem, it is used to describe the number of Statistics majors in the entire population, where each student independently has a probability p of being a major.

Hypergeometric Distribution

The hypergeometric distribution describes the probability of drawing a specific number of successes from a finite population without replacement. In the problem, although the total number of Statistics majors is binomially distributed, when sampling without replacement, the conditional number of majors in the sample follows a hypergeometric distribution, reflecting the fact that the probability changes with each draw.

Simple Random Sampling

Simple random sampling is a method of selecting a subset of individuals from a larger population such that every possible subset of the given size has an equal probability of being chosen. This method ensures unbiased representation, and its application in the problem serves to transition from the overall binomial model of majors to the hypergeometric model in the sample.

Law of Total Probability

The law of total probability is a fundamental rule that allows the calculation of an overall probability by conditioning on a partition of the sample space. In this problem, it is applied by conditioning on the possible total number of Statistics majors in the school and summing the contributions from each condition to find the probability mass function of the number of majors in the sample.

Transcript

00:01 So this problem wants you to use an appellate that is included with your textbook.

00:05 I do not have access to that, so i will be doing this problem by hand.

00:11 But regardless, you should have this formula written down somewhere, because this is how you find probabilities in a binomial situation, like in this problem, where students are either left -handed or not left -handed.

00:30 Can use our binomial equation here.

00:35 We need to know some information.

00:38 We know what n is, because that's our sample size, and we have 30 students.

00:44 We also know what our probability is.

00:47 It's 11%, but we need to convert this to a decimal.

00:52 So it's going to be 0 .11.

00:56 Now, for part a, we're going to find the probability that exactly five are left -handed.

01:05 So it's going to be 30 choose -five times our probability.

01:11 So 0 .11 raised to the 5th power times 1 minus 0 .11 raised to the 30 minus 5th power.

01:21 And you should get 0 .125.

01:25 Now for part b, we want to find the probability that it's either 4, 3, 2, 1, or 0.

01:38 So we are actually going to take all of those probabilities and add them up.

01:45 So probability that less than or equal to 4 is equal to probability of 0 plus probability of 1 plus 2 plus 3 plus 4.

02:03 So you're going to use this same equation or your applet in order to find each of these probabilities...

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