91Ó°ÊÓ

Discussion with students \(\quad\) In a statistics class of 30 stu- \(-\) dents, 20 students are from the business program and 10 students are from the science program. The instructor randomly select three students, successively and without replacement, to discuss a question. a. True or false: The probability of selecting three students from the business program is \((2 / 3) \times(2 / 3) \times(2 / 3)\). If true, explain why. If false, calculate the correct answer. b. Let \(\mathrm{A}=\) first student is from the business program and \(\mathrm{B}=\) second student is from the business program. Are A and B independent? Explain why or why not. c. Answer parts a and \(b\) if each student is replaced in the class after being selected.

Step by step solution

01

Identify Initial Conditions

We have a total of 30 students, with 20 students from the business program and 10 from the science program. Three students are chosen without replacement.

02

Evaluate Given Probability Statement

The probability of selecting three business students without replacement is not \(\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}\). Initially, the chance is \(\frac{20}{30}=\frac{2}{3}\) (first student). Next student: \(\frac{19}{29}\) as one business student is removed. Last selection: \(\frac{18}{28}\).

03

Calculate Correct Probability for Part (a)

To find the probability of picking three business students in succession without replacement, multiply these reduced probabilities: \[\frac{20}{30} \times \frac{19}{29} \times \frac{18}{28} = \frac{1140}{2436}\].

04

Check Independence of Events in Part (b)

Events A and B are not independent because the selection of the first student influences the availability of students for the second selection. The probability changes depending on the outcome of the first pick.

05

Evaluate Conditions With Replacement

When each student is replaced after being selected, the conditions reset to the original composition of 30 students. Hence, each selection becomes independent.

06

Determine Probability With Replacement (Part c)

With replacement, the probability remains \(\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{27}\) for picking students from the business program.

07

Confirm Independence With Replacement (Part c)

Since students are replaced, each event does not affect the others, making A and B independent because the probability of getting a business student remains constant at \(\frac{2}{3}\) for each pick.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events

In probability, when we say events are independent, we mean that the outcome of one event does not impact the probability of the other event occurring. Consider the scenario where you're drawing names from a hat. If after each draw you place the name back in the hat, each event remains independent. Why? Because the starting conditions are the same for each draw, with the same number of total names in the hat, maintaining consistent probabilities for each selection.
For instance:

• If drawing names from a hat with 30 names and 20 of them are business students, your probability of drawing a business student is 20/30 for each draw if you replace the name before drawing again.
• This consistent probability, unaffected by previous draws, means the events are independent.

Independence is a crucial factor when analyzing chances, especially in situations involving multiple events as each occurrence should not alter the probability conditions.

Replacement in Probability

Replacement in probability refers to the act of putting an element back into the original set after it has been selected. This action is important because it preserves the original conditions of the set, ensuring that each draw from the set is independently distributed. Think of it analogous to resetting a board game after each turn.

• When replacement occurs, the composition of the set remains unchanged. Thus, probabilities for each potential outcome remain constant between draws.
• For example, if you draw a student from a group of 20 business and 10 science students, and then put them back each time, you are drawing with replacement. This means the chance of drawing a business student remains 20/30 for each draw.

Replacement simplifies calculations as it keeps conditions uniform across trials, making it easy to predict and verify outcomes mathematically.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. It is a fundamental concept used to determine how certain events either lead to or exclude the possibility of others.
To illustrate, consider the classroom example:

• If a business student is selected first without replacement, the likelihood of selecting another business student on the second draw changes because there are fewer business students remaining in the pool.
• This probability adjusts from 20/30 to 19/29, assuming a business student was initially chosen.

Conditional probability 'conditions' the next event based on outcomes that have already transpired, hence affecting the probability of subsequent events.
It emphasizes the interconnected nature of different events within a probability framework, spotlighting how earlier events influence the probabilities of later ones.