Mastering Permutations, Combinations, and Probability

Algebra 2 Grades High School 43:24 Video
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Lesson Description

Learn to solve permutation, combination, and probability problems with this comprehensive lesson based on Mario's Math Tutoring's video. Practice with real-world examples and word problems.

Video Resource

Permutations, Combinations, and Probability (15 Word Problems)

Mario's Math Tutoring

Duration: 43:24
Watch on YouTube

Key Concepts

  • Permutations (nPr)
  • Combinations (nCr)
  • Factorials (n!)
  • Probability as successful outcomes / total outcomes
  • Binomial Probability

Learning Objectives

  • Students will be able to differentiate between permutations and combinations and apply the correct formula.
  • Students will be able to calculate probabilities using permutations, combinations, and factorials.
  • Students will be able to solve real-world word problems involving permutations, combinations, and probability.
  • Students will be able to apply binomial probability to solve word problems.

Educator Instructions

  • Introduction to Permutations and Combinations (10 mins)
    Begin by defining permutations and combinations. Use the example of the letters 'H', 'A', and 'T' to illustrate permutations. Explain factorials and their use in calculating permutations. Introduce the formulas for nPr and nCr.
  • Probability Fundamentals (5 mins)
    Define probability as the number of successful outcomes divided by the total possible outcomes. Emphasize that permutations and combinations help determine these numbers. Explain how probability is typically expressed as a fraction or decimal.
  • Word Problem Examples (30 mins)
    Work through a selection of word problems from the video, demonstrating how to apply permutations, combinations, and factorials to calculate probabilities. Examples include distinguishable permutations of the word 'mathematics', awarding medals in a bike race, choosing co-captains, forming committees, card probabilities and binomial probability.
  • Card Deck Probabilities(15 mins)
    Card deck probabilities using combinations. Break down the composition of a standard deck (suits, face cards, numbered cards, aces). Go over the example questions like, 'How many ways are there of being dealt a five card hand out of a standard 52 card deck if all cards are numbered?', 'What is the probability of being dealt all face cards?'
  • Complex Word Problems and Review (10 mins)
    Present additional word problems, such as the decagon diagonal problem, four-digit number constraints, salad bar combinations, defective item probabilities and free throw shooting percentages to push the students further. Review key concepts and address any remaining questions.

Interactive Exercises

  • Group Problem Solving
    Divide students into groups and assign each group a different word problem. Have them work together to solve the problem and present their solution to the class.
  • Card Probability Simulation
    Using a deck of cards, have students draw hands and calculate the probabilities of different outcomes (e.g., drawing a flush, drawing a pair). Use this simulation to verify their calculations.

Discussion Questions

  • How does the concept of order influence whether you should use a permutation or a combination?
  • In what real-world scenarios might you use probability calculations involving permutations and combinations?
  • Explain in your own words the difference between permutations and combinations. Provide a unique example for each.

Skills Developed

  • Problem-solving
  • Critical thinking
  • Mathematical reasoning
  • Application of formulas

Multiple Choice Questions

Question 1:

What is the value of 5!?

Correct Answer: 120

Question 2:

In how many ways can you arrange the letters in the word 'ALGEBRA'?

Correct Answer: 2520

Question 3:

What is the formula for nCr?

Correct Answer: n! / (r! * (n-r)!)

Question 4:

A committee of 3 people is to be formed from 5 men and 4 women. How many different committees can be formed if it must consist of 2 men and 1 woman?

Correct Answer: 40

Question 5:

What is the probability of drawing a heart from a standard deck of 52 cards?

Correct Answer: 1/4

Question 6:

What is the difference between permutation and combination?

Correct Answer: Permutation cares about order

Question 7:

If a coin is flipped 5 times, what is the probability of getting exactly 3 heads?

Correct Answer: 5/16

Question 8:

If there are 8 women and 6 men, how many ways can a committee of 4 be formed such that all are women?

Correct Answer: 70

Question 9:

You're at a salad bar with 10 different items. How many salads are possible, assuming you can choose any combination of items?

Correct Answer: 1023

Question 10:

A bag contains 5 red marbles and 3 blue marbles. What's the probability of drawing 2 red marbles without replacement?

Correct Answer: 5/14

Fill in the Blank Questions

Question 1:

The formula for permutations, where order matters, is _______.

Correct Answer: nPr

Question 2:

The exclamation point '!' in mathematics represents a _______.

Correct Answer: factorial

Question 3:

When the order of selection does not matter, use a _______ to calculate possible outcomes.

Correct Answer: combination

Question 4:

Probability is defined as the number of successful outcomes divided by the total number of _______.

Correct Answer: outcomes

Question 5:

A decagon has _______ sides.

Correct Answer: 10

Question 6:

In a standard deck of cards, there are _______ suits.

Correct Answer: 4

Question 7:

In probability, 'or' means _______, combining possibilities.

Correct Answer: union

Question 8:

Events that cannot occur at the same time are said to be _______ exclusive.

Correct Answer: mutually

Question 9:

Choosing members for a committee, where roles aren't assigned, is an example of ______.

Correct Answer: combination

Question 10:

If a person's free throw percentage is 80 percent, then the chances of missing a free throw is ______ percent.

Correct Answer: 20

Educational Standards

A2.D.3:Use probability to evaluate outcomes of decisions. 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).

Teaching Materials

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