NCEA LEVEL 3 PROBABILITY CONCEPTS

Sample space and events

A sample space is the set of all possible outcomes in a statistical experiment. For example, {(H,H), (H,T), (T,T), (T,H)} is the sample space of possible outcomes of two throws of a coin. An event is a subset of the sample space. An event is said to occur if any of the outcomes making up that event occur.

Probability

The probability (p) of an event is a measure of how likely an event is to occur, where 0 ≥ p ≤ 1

Experimental probability

An experiment or simulation is run and the long-run proportion of times an event occurred is calculated. The greater the number of trials, the more reliable the estimate.

Theoretical probability

Based on a probability model, with the intention that the model matches the data closely. In simple cases, where outcomes in a discrete sample space are equiprobable (equally likely to occur):
P(event) = number of outcomes in event / number of outcomes in sample space
If an event has probability p of occurring during a trial, then the expected number of occurrences of the event after n independent trials is np.

What is the probability of events A and B both occurring?

A AND B (A ∩ B) is the intersection of A and B and is the set of outcomes in both A and B.
P (A ∩ B) = P(A) x P(B)

What is the probability of event A or B occurring?

A OR B (A U B) is the union of A and B and is the set of outcomes in A or B or both.
P (A U B) = P(A) + P(B) - P (A ∩ B)

What is the probability of event A not occurring?

NOT A (A') is the set of outcomes not in A.
P (A') = 1 - P(A)

Mutually exclusive events

A and B are mutually exclusive if they cannot occur together (i.e. (A ∩ B) = 0). If A and B are mutually exclusive then:
P(A ∩ B) = 0
P (A U B) = P(A) + P(B)

Independent events

Two events are independent if the occurrence of one event is not affected by the occurrence of the other event. If events A and B are independent, then:
P (A ∩ B) = P(A) x P(B)

Complementary events

Where there are 2 possible outcomes, (e.g true or not true). The complementary events always adds to 1.
Events A and B are said to be complementary if they are the only 2 possible outcomes:
P (A U B) = 1
P(A ∩ B) = 0
A is the complement of B (B = A')

Conditional probability

A conditional probability is calculated using a reduced sample space, as other events have already occurred. This formula gives the probability of an event A occurring given that event B has already occurred:
P (A | B) = P(A ∩ B) / P(B)
This can be rearranged to:
P(A ∩ B) = P(A) x P (B | A)
P(A ∩ B) = P(B) x P (A | B)

Permutations

A permutation is an ordered arrangement. For example, there are 6 different permutations of the letters A, B, C:
ABC, ACB, BAC, BCA, CAB, CBA
The number of permutations of n distinct objects is called n factorial (n!).
n! = n x (n-1) x (n-2) x ... x 3 x 2 x 1
For example, 3 objects can be arranged in a line in 3! = 3 x 2 x 1 = 6 ways.
(n + 1)! = (n + 1) x n!
(n - 1)! = n! / n
The number of permutations of r objects selected from a group of n distinct objects is:
nPr = n! / (n - r)!

Combinations

A combination is an unordered selection (order is of no importance). For example, there are 3 possible selections of 2 letters from A, B, C:
- A and B are selected
- A and C are selected
- B and C are selected
The number of combinations of r objects from n distinct objects is written nCr where:
nCr = nPr / r! = n! / ((n - r)! x r!)

Experiment

A process that produces the result(s) by chance.

Trial

A single event producing a result by chance.

Sample Space

A list (or set) of all possible outcomes

Relative Frequency

The frequency of a result divided by the frequency of all possible results. The relative frequency is also called the experimental probability.

Equally likely outcomes

Where all outcomes have the same probability of occurring.

Tree Diagram

A process that produces the result(s) by chance

Node

A single event producing a result by chance

Outcome

The frequency of a result divided by the frequency of all possible results.