Mathematics 01: Probability Foundations

Probability answers one question:

How likely is an event to happen?

A probability is always between 0 and 1.

• 0 means impossible.
• 1 means certain.
• 0.5 means a 50% chance.

1. The words you need

Question 1 — What is the chance of rolling an even number?

A fair die has six equally likely outcomes: {1, 2, 3, 4, 5, 6}.

The favourable outcomes for an even number are {2, 4, 6}. There are three.

Formula

P(A) = number of favourable outcomes / total number of equally likely outcomes

Answer

P(even) = 3 / 6 = 1 / 2 = 0.5

There is a 50% chance of rolling an even number.

This simple fraction formula works only when every outcome has the same chance. A fair die and a well-shuffled deck are common examples.

2. The three basic probability rules

Rule 1 — Range rule

0 ≤ P(A) ≤ 1

A probability cannot be negative and cannot be greater than 1.

Rule 2 — Certain event rule

P(Ω) = 1

Something in the full sample space must happen. When rolling a die, you must get one of {1, 2, 3, 4, 5, 6}.

Rule 3 — Impossible event rule

P(∅) = 0

∅ means an empty event. Rolling a 7 on an ordinary six-sided die is impossible.

Question 2 — What is the chance of not rolling a 6?

Let A mean “roll a 6.” Its complement, written Aᶜ, means “do not roll a 6.”

Complement formula

P(Aᶜ) = 1 - P(A)

Answer

P(roll 6) = 1 / 6
P(not 6) = 1 - 1 / 6 = 5 / 6 ≈ 0.83

There is an 83% chance of not rolling a 6.

Question 3 — What is the chance of drawing a heart or a king?

A standard deck has 52 cards.

• 13 cards are hearts.
• 4 cards are kings.
• 1 card is both a heart and a king: the king of hearts.

If we simply add 13 + 4, we count the king of hearts twice. We must subtract it once.

Addition rule

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Answer

P(heart or king) = 13/52 + 4/52 - 1/52
                  = 16/52
                  = 4/13 ≈ 0.31

There is about a 31% chance of drawing a heart or a king.

Key formula summary

Basic probability:     P(A) = favourable / total
Complement:            P(Aᶜ) = 1 - P(A)
Addition rule:         P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Range:                 0 ≤ P(A) ≤ 1
Certain event:         P(Ω) = 1
Impossible event:      P(∅) = 0

Practice

1. A bag has 3 red and 7 blue balls. What is P(red)?
2. For the same bag, what is P(not red)?
3. In a deck, what is P(queen or spade)? Remember the queen of spades belongs to both groups.

Answers

1. 3/10 = 0.30
2. 7/10 = 0.70
3. 4/52 + 13/52 - 1/52 = 16/52 = 4/13