Definition 1: Typically in the field of statistics we study data that results from experiments. An experiment can be considered to be a series of trials, each with a particular outcome. An event is a collection of outcomes corresponding to some result in the experiment. The number of outcomes in event E (i.e. the number of elements in set E) is written as |E|. The set of all possible outcomes is called the sample space, often designed  S. An event is then simply a subset of the sample space. The probability P(E) of the event E is |E| / |S|, assuming S is not empty.

Example 1: Consider the simple experiment of tossing a coin twice. What is the probability that the coin comes up heads both time?

The sample space S = {HH, HT, TH, TT} and the required event E = {HH}. Thus the probability that the coin is heads both times is P(E) = |E| / |S| = ¼, or 25%.

Observation: We now state the fundamental properties of probability, using the usual set notation.

Property 1:

1. 0 ≤ P(A) ≤ 1
2. P(Ø) = 0
3. P(S) = 1
4. P(A′) = 1 – P(A), where A′ = S – A
5. P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Proof: Simple consequences of Definition 1.

Example 2: Consider the experiment of drawing one card from a standard deck of 52 cards. What is the probability of drawing either a spade or face card?

There are 13 spades and 12 face cards, but 3 of these face cards are also spades, which we should not count twice. Thus, there are 13 spades and 9 non-spade face cards for a total of 22 cards out of 52. The probability is therefore 22/52. We now show how to calculate the result using Property 1e.

Let A = the event that a spade is drawn and B = the event that a face card (King, Queen or Jack) is drawn. P(A) = 13/52, P(B) = 12/52 and P(A ∩ B)= 3/52. Thus the probability of drawing either a spade or face card is P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = 13/52 + 12/52 – 3/52 = 22/52.

Definition 2: The probability that an event A occurs assuming that event B occurs is called the conditional probability of A given B and is denoted P(A|B).

Observation: By Definitions 1 and 2

assuming if B ≠ Ø.

Property 2:

1. P(A|B) ∙ P(B) = P(A∩B) = P(B|A) ∙ P(A)
2. P(A|B) = P(B|A) ∙ P(A) / P(B) called Bayes’ Theorem
3. P(A) = P(A|B) ∙ P(B) + P(A|B′) ∙ P(B′) called the Law of Total Probability

Proof: The first assertion is a restatement of the last observation. The second assertion is a consequence of two applications of the first since

We now prove the third assertion. Since A  = (A∩B) ∪ (A∩B′), by Properties 1b and 1e,

Now by Property 2a and 2b,

which proves the third assertion.

Example 3: Consider the experiment of picking two balls at random without replacement from a bag which contains 3 reds and 2 blacks. What is the probability that both balls are red?

Let A = a red ball is taken on the first draw and B = a red ball is taken on the second draw. The probability that the first draw is red is P(A) = 3/5. The probability that the second draw is red given the first draw is red is P(B|A) = 2/4 = ½. From Property 2a, we see that the probability that both draws are red is

Definition 3: Two events A and B are independent if P(A∩B) = P(A) ∙ P(B)

Property 3: Two events A and B are independent if and only if P(A) = P(A|B)

Proof: A and B are independent if and only if P(A∩B) = P(A) ∙ P(B), which by Property 2a is true if and only if P(A|B) ∙ P(B) = P(A) ∙ P(B), which in turn is true if and only if P(A|B) = P(A).

Observation: A and B are independent if B’s occurring (or not occurring) has no influence on A’s occurring, i.e. it doesn’t increase or decrease the probability of A occurring. By Property 3, A and B are independent if any only if P(B|A) = P(B), and so it also follows that if A and B are independent then A’s occurring has no influence on B’s occurring either.

Example 4: Repeat the experiment from Example 3, but this time we put the ball picked on the first draw back in the bag before drawing a second ball (i.e. sampling with replacement).

Since P(B|A) = 3/5 = P(B), A and B are independent, it follows that

P(A∩B) = P(A) ∙ P(B) = 3/5 ∙ 3/5 = 36%.

Example 5: You have two bags, one containing 3 red and 2 black balls, the other containing 1 red, 1 blue and 2 black balls. You pick a bag at random and then pick a ball from that bag at random. What is the probability that the ball picked is red?

Let A = event that the first bag is picked and let B = event that a red ball is drawn. By Property 2c,

P(B) = P(B|A) ∙ P(A) + P(B|A′) ∙ P(A′) = .6(.5) + .25(.5) = 42.5%.

Example 6: Suppose you role a die 12 times. What is the probability that the number 1 will not appear on any of the throws? What is the probability that the number 1 will appear on at least one of the 12 throws?

The 12 throws represent 12 independent events. The probability of throwing a 1 on any single trial is 1/6 and so the probability of not throwing a 1 on any single trial is 1 – 1/6 = 5/6 (by Property 1d). Thus the probability of not throwing a 1 on any of the 12 throws is (5/6)¹² = 11.2% (by Definition 3).

The probability that the number 1 will appear at least once is simply 1 – 11.2% = 88.8% (by Property 1d). This is equivalent to 1 – (1 – 1/6)¹².