Sample Space (S) is the set of ALL POSSIBLE OUTCOMES

of a random experiment

• Written in curly braces { }

• Lists every possible result

Event: A subset of the sample space

Example: Rolling an even number = {2, 4, 6}

If one event can occur in m ways

and another event can occur in n ways,

then BOTH events can occur in m × n ways

Total Outcomes = n₁ × n₂ × n₃ × ...

n₁ = number of ways for first event

n₂ = number of ways for second event

and so on...

Problem: A restaurant offers 3 types of sandwiches and 4 types of drinks. How many meal combinations?

Problem: Flip a coin AND roll a die. How many outcomes?

Probability measures the LIKELIHOOD

that an event will occur

• Probability is always between 0 and 1

• Can be written as fraction, decimal, or percent

P(A) = n(A) ÷ n(S)

or

P(Event) = Favorable Outcomes ÷ Total Outcomes

P = 0: Impossible (will NEVER happen)

P = 0.5: Equally likely (50-50 chance)

P = 1: Certain (will ALWAYS happen)

Problem: What is the probability of rolling a 4 on a fair die?

Use probability to predict

EXPECTED OUTCOMES

when an experiment is repeated many times

Expected = Probability × Number of Trials

or

E = P × n

Problem: If you flip a coin 100 times, how many times would you expect to get heads?

Problem: If you roll a die 60 times, how many times would you expect to roll a 5?

Complementary events are OPPOSITE events

If one event happens, the other CANNOT happen

• Notation: A' or Aͨ (read as "A complement")

• Together they cover ALL possible outcomes

P(A) + P(A') = 1

or

P(A') = 1 − P(A)

Problem: The probability of rain tomorrow is 0.3. What is the probability of NO rain?

✓ Sample space = ALL possible outcomes in { }

✓ Counting principle: MULTIPLY to find total outcomes

✓ Probability formula: P(A) = Favorable ÷ Total

✓ Probability range: Always 0 ≤ P ≤ 1

✓ P = 0 means impossible, P = 1 means certain

✓ Predictions: Expected = Probability × Number of trials

✓ Complementary events: P(A) + P(A') = 1

✓ Complement formula: P(not A) = 1 − P(A)

✓ Simplify fractions in probability answers

✓ Can write probabilities as fractions, decimals, or percents

Sample Space:

"Sample Space is the SET, of outcomes we can get!"

Counting Principle:

"When events are combined, just MULTIPLY - that's how you find all outcomes nearby!"

Probability:

"What you WANT over what you've GOT - favorable over total, that's the plot!"

Predictions:

"Probability times trials, gives expected - predictions made, well connected!"

Complement:

"One minus P(A) gives P(A prime) - the opposite event, working just fine!"

P = 0 or P = 1:

"Zero is NEVER, One is SURE - probabilities in between are less secure!"