Dicreet mercy final

What is the Multiplication Principle of Counting?

If one event can occur in n₁ ways and another independent event can occur in n₂ ways, both can occur in n₁ × n₂ ways.

Formula for Multiplication Principle of Counting

Total Outcomes = n₁ × n₂ × n₃ × …

What condition is required for the Multiplication Principle?

Events must be independent.

If there are 3 entrées, 4 fruits, and 2 drinks, how many possible meals are there?

24

If a student chooses from 5 shirts, 3 pants, and 2 shoes, how many outfits are possible?

30

What is a permutation?

An arrangement of objects where order matters.

Examples of permutations

Ranking contestants, arranging books, seat assignments.

Formula for permutations

P(n,r) = n! / (n-r)!

What does n represent in permutations?

Total number of objects.

What does r represent in permutations?

Number of objects chosen.

What is factorial?

The product of all positive integers from a number down to 1.

What is 5!?

120

What is 4!?

24

What is 3!?

6

What is 0!?

1

How many ways can 3 students be arranged from 5 students?

60

What is a permutation with repeated objects?

An arrangement where some objects are identical.

Formula for permutations with repeated objects

n! / (n₁! × n₂! × … × nᵣ!)

How many arrangements are possible for the letters in BALLOON?

1260

Formula for arranging all objects

P(n,n) = n!

How many ways can 5 books be arranged?

120

What is a combination?

A selection of objects where order does not matter.

Examples of combinations

Selecting committee members, choosing winners, picking team members.

Formula for combinations

C(n,r) = n! / [r!(n-r)!]

Main difference between permutation and combination

Permutation: order matters; Combination: order does not matter.

How many ways can 2 students be chosen from 20 students?

190

If order matters, should you use permutation or combination?

Permutation.

If order does not matter, should you use permutation or combination?

Combination.

What is probability?

A measure of the likelihood that an event will occur.

What is theoretical probability?

Probability based on reasoning and expected outcomes.

Probability of getting heads on a fair coin toss

1/2

What is experimental probability?

Probability based on actual observations or collected data.

If a coin lands heads 60 times in 100 tosses, what is the experimental probability of heads?

0.60

What is an experiment in probability?

An activity involving chance.

Examples of probability experiments

Tossing a coin, rolling a die.

What is a sample space?

The set of all possible outcomes.

Sample space of a coin toss

{H, T}

Sample space of a die roll

{1, 2, 3, 4, 5, 6}

What is an event?

One or more outcomes of an experiment.

Examples of events

Getting heads, rolling an even number.

Formula for probability

P(E) = n(E) / n(S)

What does n(E) represent?

Number of favorable outcomes.

What does n(S) represent?

Total number of outcomes.

Probability of rolling a 4 on a fair die

1/6

What is the complement rule?

The probability that an event does not occur.

Formula for complement rule

P(E') = 1 - P(E)

Probability of NOT rolling a 6 on a fair die

5/6

What are mutually exclusive events?

Events that cannot happen at the same time.

Example of mutually exclusive events

Rolling a 2 and rolling a 5 on one die roll.

Addition rule for mutually exclusive events

P(A or B) = P(A) + P(B)

Probability of rolling a 2 or 5 on a die

1/3

What are not mutually exclusive events?

Events that can happen together.

Addition rule for not mutually exclusive events

P(A or B) = P(A) + P(B) - P(A and B)

In a class of 40 students, 20 play basketball, 15 play volleyball, and 5 play both. What is P(Basketball or Volleyball)?

0.75

What are independent events?

Events where one event does not affect the other.

Examples of independent events

Coin tosses, rolling dice.

Multiplication rule for independent events

P(A and B) = P(A) × P(B)

Probability of getting heads then heads

1/4

What does "with replacement" indicate?

Independent events.

Probability of drawing a red sock then another red sock with replacement if 1 out of 5 socks is red

1/25 or 4%

What are dependent events?

Events where the first event affects the second.

Example of dependent events

Drawing cards without replacement.

What does "without replacement" indicate?

Dependent events.

If a bag has 6 black pens and 8 blue pens, what is the probability of drawing two black pens without replacement?

15/91 or about 16.48%

What is conditional probability?

The probability that event B occurs given that event A has already occurred.

Notation for conditional probability

P(B|A)

How do you read P(B|A)?

Probability of B given A.

Exam tip: What operation is usually used for "AND"?

Multiply.

Exam tip: What operation is usually used for "OR"?

Add.

Exam tip: What rule is useful for NOT questions?

Complement Rule.

Exam tip before solving permutation or combination problems

Identify n and r first.