Three students are selected at random from a group of 3...

Three students are selected at random from a group of 3 sophomores and 3 juniors. The table and relative-frequency histogram show the distribution of the number of sophomores chosen. Find each probability. $\begin{array}{|c|c|c|c|}\hline 0 & {1} & {2} & {3} \\ \hline 1 & {\frac{9}{20}} & {\frac{9}{20}} & {\frac{1}{20}} \\ \hline\end{array}$ P(2 sophomores)

Three students are selected at random from a group of 3 sophomores and 3 juniors. The table and relative-frequency histogram show the distribution of the number of sophomores chosen. Find each probability.
$\begin{array}{|c|c|c|c|}\hline 0 & {1} & {2} & {3} \\ \hline 1 & {\frac{9}{20}} & {\frac{9}{20}} & {\frac{1}{20}} \\ \hline\end{array}$

P(2 sophomores)

Key Concepts

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways events can occur, often using combinations (binomial coefficients) to determine both the total number of outcomes and the number of favorable outcomes. This fundamental tool in probability helps in analyzing situations where items or individuals are chosen from a set without regard to order.

Hypergeometric Distribution

The hypergeometric distribution is used to model scenarios in which samples are drawn without replacement from a finite population containing two types of objects (often labeled as successes and failures). It provides the probability of drawing a specific number of successes in a fixed number of draws, making it central to problems where the probability changes on each draw due to the decreasing population.

Probability Distribution

A probability distribution describes how probabilities are assigned to the different outcomes of a random experiment. It represents the likelihood of each possible event occurring and is a key concept in understanding how random variables behave, whether the distribution is provided by combinatorial reasoning or observed through relative frequencies.

Relative Frequency

Relative frequency is the ratio of the number of times an event occurs to the total number of trials or observations. In probability, it can represent the empirical probability of an event and is frequently used to validate theoretical probabilities or illustrate a probability distribution through graphical representations such as histograms.

Transcript

00:01 All right, we need the probability of two sophomores.

00:02 And if you watch the previous video, i'm not going to explain this super thoroughly.

00:05 You could just look at the histogram and figure out that that probability is there.

00:11 It is right above two -fifths.

00:14 And if you know that every single little bar is counting by a 20th, you can see two -fifths is 8 .20ths, and it's just one little bar above it.

00:23 So it will be 9 .20ths.

00:27 But that's not very thorough explanation.

00:29 I'm going to give you a thorough explanation.

00:31 Probability of two sophomores.

00:32 Well, let's say the probability that the first person is a sophomore is a sophomore is three out of six because there's three sophomores out of six people.

00:39 Probably that the second person would be a sophomore is two out of five because there's only two sophomores left out of five people left.

00:47 And then the probability you'd get a junior at that last slot would be three out of the four remaining people.

00:57 And so this would be the probability as if the sophomores were in the first and second locations out of three.

01:04 Let's do some cross -canceling to make this simpler.

01:06 We can divide three and six by three to get a one and a two...

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