Probability with Combinations: Defective Widgets

Algebra 2 Grades High School 3:01 Video
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Lesson Description

Learn how to calculate probabilities using combinations, focusing on scenarios involving defective units. This lesson covers calculating the probability of selecting good units, defective units, and 'at least' scenarios.

Video Resource

Probability Defective Units Example

Mario's Math Tutoring

Duration: 3:01
Watch on YouTube

Key Concepts

  • Probability as a ratio of successful outcomes to total possible outcomes
  • Combinations (nCr) and when to use them
  • Calculating probabilities of independent events

Learning Objectives

  • Students will be able to calculate the probability of selecting a specific number of good or defective units from a larger set.
  • Students will be able to apply combinations to solve probability problems where order does not matter.
  • Students will be able to calculate probabilities for 'at least' scenarios by considering multiple possible outcomes.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of probability and combinations. Briefly discuss scenarios where combinations are used instead of permutations.
  • Video Explanation (10 mins)
    Play the Mario's Math Tutoring video 'Probability Defective Units Example'. Encourage students to take notes on the problem-solving approach and formulas used.
  • Part A: All 4 Good (10 mins)
    Walk through the first part of the video (0:35-1:31) step-by-step. Emphasize how to determine the number of good widgets and set up the combination formula. Discuss why we are choosing 4 from 10 and 0 from 5. Explain how these calculations relate to the desired outcome.
  • Part B: Exactly 2 Good (10 mins)
    Explain Part B (2:00-2:34) where exactly 2 widgets are good. Emphasize the importance of understanding 'exactly'. Make sure students understand that because there are four widgets total, if two are good, two must be defective.
  • Part C: At Least 2 Good (15 mins)
    Explain Part C (2:34-end) involving 'at least' scenarios. Stress the need to consider multiple cases (2 good, 3 good, and 4 good) and sum their probabilities. Explain that each of these probabilities are their own independent calculation which must be added together.
  • Practice Problems (15 mins)
    Provide students with similar probability problems to solve using combinations. Have them work independently or in pairs. Encourage students to show all steps of their work.
  • Review and Q&A (5 mins)
    Review the key concepts and answer any remaining questions. Emphasize the importance of identifying the total possible outcomes and the specific successful outcomes.

Interactive Exercises

  • Widget Simulation
    Create a virtual simulation where students can repeatedly draw 4 widgets from a set of 15 (5 defective). Have them track the number of times they get all good, exactly 2 good, and at least 2 good. Compare the experimental probabilities to the calculated probabilities.
  • Card Drawing
    Use a standard deck of cards to explore similar probability problems involving combinations. For example, 'What is the probability of drawing exactly 2 aces in a hand of 5 cards?'

Discussion Questions

  • When is it appropriate to use combinations instead of permutations in probability problems?
  • How does the phrase 'at least' affect the way you set up a probability calculation?
  • Can you think of other real-world scenarios where calculating probabilities with combinations would be useful?

Skills Developed

  • Problem-solving
  • Critical thinking
  • Probability calculation
  • Application of combinations

Multiple Choice Questions

Question 1:

A company sells 20 items, 6 of which are defective. What is the probability that if you purchase 3 items, all 3 are good?

Correct Answer: (6C0 * 14C3) / (20C3)

Question 2:

In a batch of 12 items, 4 are known to be faulty. If you randomly select 2 items, what is the probability that exactly one is faulty?

Correct Answer: (4C1 * 8C1) / (12C2)

Question 3:

A box contains 8 red balls and 5 blue balls. If 3 balls are drawn at random, what is the probability that at least 2 are red?

Correct Answer: (8C2 * 5C1 + 8C3) / (13C3)

Question 4:

Which formula represents the total possible outcomes of selecting 5 objects out of 18?

Correct Answer: 18C5

Question 5:

What does 'nCr' represent in probability calculations?

Correct Answer: Combination

Question 6:

If 'at least 3' items must be selected, what scenarios do you need to calculate the probability of?

Correct Answer: All of the above

Question 7:

In a group of 10 people, 4 are chosen for a committee. If you are one of the 10 people, what calculates the probability of you being on the committee?

Correct Answer: 4/10

Question 8:

Why do we use combinations instead of permutations when selecting widgets?

Correct Answer: Because the order of the widgets does not matter

Question 9:

You have 15 widgets and wish to pick 4. How would you find the total possible outcomes?

Correct Answer: 15C4

Question 10:

What does it mean for an item to be defective?

Correct Answer: Item does not work as intended

Fill in the Blank Questions

Question 1:

When calculating probability, the denominator represents the _____ possible outcomes.

Correct Answer: total

Question 2:

The formula nCr is used when the ______ of selection does not matter.

Correct Answer: order

Question 3:

If a problem asks for the probability of 'at least' two successes, you must consider the cases where you have two, three, or more ________.

Correct Answer: successes

Question 4:

In the video example, if there are 15 total widgets and 5 are defective, then there are ____ good widgets.

Correct Answer: 10

Question 5:

When calculating the probability of selecting exactly 'x' good items, you also need to consider how many ________ items are selected.

Correct Answer: defective

Question 6:

A company selling 20 widgets, requires you to pick 4. The number of ways this can be accomplished is represented as _____.

Correct Answer: 20C4

Question 7:

If 3/20 widgets are defective, the probability that a randomly chosen widget is not defective is ______.

Correct Answer: 17/20

Question 8:

A situation in which you have to select between 3 items requires ______.

Correct Answer: 3C3

Question 9:

Defective items do not work ______.

Correct Answer: correctly

Question 10:

Probability is calculated by dividing what you want by the ______ possible outcomes.

Correct Answer: total

Educational Standards

A2.D.3.2:Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). A2.D.3.1:Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

Teaching Materials

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