Conditional probability refresher

Probability is always…

A number between 1 (certain) and 0 (impossible):

1 ≤ P ≥ 0

The probability experiment

A process that can be repeated & in which the outcome will be uncertain

rolling a dice, tossing a coin, selecting a card

The sample space

A list of all possible outcomes of a probability experiment, written in set notation which is denoted by Ω or S

rolling a dice    Ω = {1, 2, 3, 4, 5, 6}
tossing a coin   Ω = {H, T}

Note how tossing 2 coins can result in different sample space scenarios, depending on whether you care about the order or not

coin toss, order important        Ω = {HH, HT, TH, TT}
coin toss, order un important  Ω = {HH, HT, TT}

Discrete vs Continuous sample spaces

Discrete – I can count it, even if there are a LOT of values

Shoe sizes are discrete – I could be a 6 or a 6.5 or a 7… there are only so many options

Continuous – I can’t count it, it could be anything

Foot sizes are continuous – my foot length could be 24cm, but the next person might be 24.1 and the next 24.10456 → ∞

Events

An event is a subset of a sample space – usually represented by some capital letter. There are many different types of events:

Elementary event

One item selected from a sample space, calculated by:

P(A) = 1 / Ω

Example: a dice is thrown once, what is the probability of a 6?
P(A) = 1/6 or 0.167

Impossible event

No items from the sample space, probability = 0!

Certain event

All items from the sample space, probability = 1!

Independent events

Two or more events, where the outcome of the 1st event will not affect the outcome of the next event, and so on – think “and”, which is calculated by:

P(A∩B) = P(A) x P(B)

Example: a card is selected from a deck of 52 cards, it is then put back in the deck and a second card is selected. What is the probability of choosing a king and then a nine?
P(A – king) = 4 / 52
P(B – nine) = 4 / 52
P(A∩B) = (4 / 52) x (4 / 52) = 1 / 169 = 0.006

Dependent events

Two or more events, where the outcome of the 1st event affects the outcome of the next event, and so on – think “and”, which is calculated by:

P(A∩B) = P(A) x P(B|A)

Example: a card is selected from a deck of 52 cards, it is kept to one side and a second card is selected. What is the probability of choosing a king and then a nine?
P(A – king) = 4 / 52
P(B|A – nine, given a king was already drawn) = 4 / 51
P(A∩B) = (4 / 52) x (4 / 51) = 4 / 663 = 0.006

Mutually exclusive events

Two or more events, where the outcome cannot be can only be one of the identified events – think “or”, which is calculated by:

P(A∪B) = P(A) + P(B)

Example: a dice is thrown once, what is the probability of a 2 or a 5? (Clearly, in a single role of a dice (die??) you could never get both a 2 and a 5!)
P(A – throwing a 2) = 1 / 6
P(B – throwing a 5) = 1 / 6
P(A∪B) = 1 / 6 + 1 / 6 = 1 / 3 = 0.333

Non-mutually exclusive events

Two or more events, where the outcome of the first may have an impact on the second – think “or”, which is calculated by:

P(A∪B) = P(A) + P(B) – P(A∩B)

Example: a card is selected from a deck of 52 cards, what is the probability of choosing a king or a club? (What we want to avoid here is counting the probability of the king of clubs twice!)
P(A – drawing a king) = 4 / 52
P(B – drawing a club) = 13 / 52
P(A∩B) = 1 / 52
P(A∪B) = 4 / 52 + 13 / 52 – 1 / 52 = 4 / 13 = 0.308