Unit 8: Problem Solving · Lesson 7
Problem Solving - Lesson 7
What this lesson is for
- Students will understand and practice Problem Solving - Lesson 7
- Students will solve problems related to this topic
Train up a child in the way he should go: and when he is old, he will not depart from it.
Proverbs 22:6
Counting and Probability Problems
"The lot is cast into the lap; but the whole disposing thereof is of the LORD." — Proverbs 16:33 (KJV)
Even the casting of lots — the ancient equivalent of drawing straws — is under God's sovereign governance. Probability quantifies uncertainty; combinatorics counts possibilities. Together they form the mathematical framework for every decision made under uncertainty.
Key Concepts
Fundamental Counting Principle: if event A can happen in m ways and B in n ways, both together happen in m*n ways.
Permutations (order matters): P(n, r) = n! / (n-r)!
Combinations (order does not matter): C(n, r) = n! / (r! * (n-r)!)
Probability (equally likely outcomes): P(E) = (favorable outcomes) / (total outcomes).
- Complement rule: P(not E) = 1 - P(E).
- Addition rule: P(A or B) = P(A) + P(B) - P(A and B).
- Multiplication rule (independent events): P(A and B) = P(A) * P(B).
- Conditional probability: P(A|B) = P(A and B) / P(B).
Worked Example
A committee of 3 is chosen from 8 people. What is the probability that a specific person is included?
Total committees: C(8,3) = 56. Committees with the specific person: C(7,2) = 21. P = 21/56 = 3/8.
4-letter arrangements from MATH (no repeats): P(4,4) = 4! = 24.
Practice Problems
- A restaurant offers 4 soups, 6 entrees, and 3 desserts. How many different complete meals can be ordered?
- From a class of 15, how many ways can a president, vice president, and secretary be chosen?
- A bag has 5 red and 3 blue marbles. Find the probability of drawing 2 red in a row without replacement.
- A test has 10 true/false questions. If a student guesses randomly, find the probability of scoring at least 9 correct.
Think About It: Proverbs says God disposes the outcome of the lot — He is sovereign over events we call random. How should understanding God's sovereignty over all outcomes affect your attitude toward uncertainty, risk, and decision-making in your own life?
Guided practice
A committee of 2 people is chosen from 5 people. In how many ways can this be done? (Use combination formula since order does not matter: C(n,r) = n!/(r!(n-r)!).)
Show the steps and the answer
C(5,2) = 5!/(2!*3!) = (5*4)/(2*1) = 20/2 = 10.
Answer: 10
How many 3-letter arrangements can be made from the letters in GARDEN (no repeats)? Use the permutation formula P(n,r) = n!/(n-r)!
Show the steps and the answer
P(6,3) = 6!/(6-3)! = 6!/3! = (6*5*4*3*2*1)/(3*2*1) = 6*5*4 = 120.
Answer: 120
A jar has 4 red marbles, 5 blue marbles, and 6 green marbles. If one marble is drawn at random, what is the probability it is blue?
Show the steps and the answer
Total marbles = 4 + 5 + 6 = 15. Blue marbles = 5. Probability = 5/15 = 1/3.
Answer: 1/3
Homework
Complete the practice problems for Problem Solving - Lesson 7.
Note for the parent
Today's lesson covers Problem Solving - Lesson 7. Estimated time: 55 minutes. Encourage independent work but be available for questions.
The first three of this lesson's 10 questions
Total committees C(8,3) = 56. Committees including a specific person: C(7,2) = 21. P(specific person included) = ?
Show the answer
Answer: 21/56 = 3/8
Fundamental Counting Principle: 4 shirt colors and 3 pant styles. Total outfits = ?
Show the answer
Answer: 12
P(A and B) for independent events A and B with P(A)=0.4, P(B)=0.3:
Show the answer
Answer: 0.12
Another 7 questions follow in the lesson itself, each one graded the moment it is answered.