PROBABILITY
Sample space
The set of all possible outcomes. E.g. the sample space of a six-sided die being rolled is A =
{1,2,3,4,5,6}
Event
The subset of the sample that’s of interest. E.g. rolling an even number when a die is rolled
= {2,4,6}
The probability of event A happening is P(A)
The number of outcomes in event A is n(A) = |A|
Revision
- In a random experiment, each outcome is equally likely
- The sample space is the set of all possible outcomes
- An event is a set of one or more outcomes
- P(E) = n(E)
n(S)
P(E) = number of desired outcomes / total number of outcomes
Express probability as a fraction. Do not round decimals as your answer
Theoretical vs experimental probability
Complementary events
- Complement of Set A is the set of all elements NOT in set A = Ā, A’
- The complement of event E is the event “not E” which can be denoted as Ē, E’ or Ec
- The sum of all possible outcomes in an experiment is 1
- P(E) + P(Ē) = 1
Mutually exclusive events
- Two events which cannot occur at the same time
- In a Venn diagram, the two sets do not intersect
- P (A ∪ B) = 0
- P (A ∪ B) = P(A) + P(B)
Independent events
- The probability of one event occurring is unchanged when the other event occurs
- P (A ∩ B) = P(A) x P(B)
Multistage experiments
- Use an array or a tree diagram to show the sample space
- Array: show outcomes of the first stage vertically and the other horizontally
Heads
Tails
Heads
Tails
-
Tree: list the outcomes of the first stage vertically, and then from each outcome list
the outcomes of the second stage
o Use when the sample space is large, and each outcome is not equally likely.
Write
probability of each branch on the branch
Conditional probability
- P (A | B) means the probability of A, given B:
o P (A | B) = P (A ∩ B)
P (B)
- If A and B are independent events, then P (A | B) = P(A)
Random variables
- A random variable uses numbers to describe the possible outcomes of a random
experiment
- A discrete random variable has values which can be listed
o A sample of 1000 people were asked how many children they have
- A continuous random variable can take any value within a given range
o A sample of 100 students were asked how long they spent studying for a
maths test
- X = random variable; x = the possible values of the random variables
Discrete probability distributions
-
Lists the probability for each value of the discrete random variable
Discrete probability function: P (X=x) or p(x)
Probabilities all add to one
All possible values for X must be mutually exclusive
Uniform distribution: all possible values for the discrete random variable have the
same probability of occurring
Mean and Expected value
- The expected value, or expectation of a random variable is the centre of the
distribution
- Mean: 𝑥 = µ = E(X) =∑𝑥p(𝑥)
Variance and standard deviation
- Measures of spread
- Standard deviation: average distance of each score from the mean (s)/(𝛔)
- Variance: square of the standard deviation 𝛔2
- Var(X) = ∑ (𝑥 - µ)2 p(𝑥) = E (X - µ)2 = E (X2) - µ2
Sample
- A census collects data on the entire population
- The mean for the population is µ and the standard deviation is 𝛔
Formula Sheet:
P (A ∩ B) = P(A) x P(B)
P (A ∪ B) = P(A) + P(B) - P (A ∩ B)
P (A | B) = P (A ∩ B)
P (B)
E(X) = µ
Var(X) = E (X - µ)2 = E (X2) - µ2
Textbook Work:
Cambridge: 12A, 12B, 12D, 12E, 12F, 12G, 13A Q1-16, 13B, 13C
Maths in Focus: 9.02, 9.03, 9.04, 9.05, 9.06, 9.07, 12.01, 12.02, 12.03, 12.04