Probability – a value between 0 and 1, inclusive, describing the relative possibility (chance or likelihood) an event will occur. It is frequently expressed as
a decimal. Three key words are used in the study of probability: experiment, outcome and event. Experiment is the process that leads to the occurrence
of one and only one of several possible observations. Outcome is the particular result of an experiment. Event is a collection of one or more outcomes of
Approaches to Probability – objective and subjective viewpoints
Objective probability is subdivided into classical probability and empirical probability.
• Classical probability is based on the assumption that the outcomes of an experiment are equally likely. Using this, the probability of an event
happening is computed by dividing the number of favourable outcomes by the number of possible outcomes.
probabilityof event happening= nuberof favourableoutcomes
• Mutually exclusive is the occurrence of one event means that none of the other events can occur at the same time.
• The variable ‘gender’ presents two mutually exclusive outcomes, male and female. Someone cannot be both male and female. A manufactured part
is defective or non-defective. It can’t be both.
• If an experiment has a set of events that includes every possible outcome, such as the events ‘an even number’ and ‘an odd number’ in the
die=tossing experiment, then the set of events is collectively exhaustive.
• Collectively exhaustive means at least one of the events must occur when an experiment is conducted.
• The classical approach can be applied to lotteries.
Empirical Probability or Relative Frequency
• This is based on the number of times an event occurs as a proportion of a known number of trials.
Numberof ×event occurred∈the past
• Probability of event happening = totalnumber of observations = percentage
• This approach is based on what is called the law of large numbers. The key to establishing probabilities empirically is that more observations will
provide a more accurate estimate of the probability.
• Law of large numbers: over a large number of trials, the empirical probability of an event will approach its true probability.
• This is the likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available.
• This is used when there is little or no experience or information on which to base a probability. This means evaluating the available opinions and
other information and then estimating or assigning the probability.
• This is like when you’re estimating the next provincial elections, likelihood you’ll get married before 30, and stuff like that
Principles of Counting – multiplication formula, permutation formula, and combination formula
• Literally multiplication of x amount of variables. Total possible number of arrangements = (m)(n) • Example: Pretend you’re going out to buy a new car. So you go to a dealer, and you see that there are three models of cars. And there are two
types of wheel covers. So you figure out the total possible number of arrangements, m is the number of models, n is the wheel cover, so you: (3)
(2)=6. There are 6 possible arrangements.
• THIS ONE YOU CAN MULTIPLY DEPENDENT VARIABLES WITH INDEPENDENT ONES.
• This is applied to find the possible number of arrangements when there is only on group of object. How many ways an event can happen.
Like how many ways can 15 students form groups of 3.
• Example: three electronic parts are to be assembled into a plug-in unit for a television set. The parts can be assembled in any order. The question
involving counting is: In how many different ways can the three parts be assembled? One order might be: the transistor first, the LEDs second, and
the synthesizer third. This arrangement is called a permutation.
• PERMUTATION: any arrangement of r objects selected from a single group of n possible objects when order is considered.
• Note that the arrangements a b c and b a c are different permutations. The formula to count the total number of different permutations is:
• N is the total number of objects, r is the number of objects selected. The ! is a factorial notation, For instance 5!= 1*2*3*4*5=120.
• To solve the example that was provided up there, so how many different ways can they be assembled?
o There are three electronic parts to be assembled, so n=3. Because all three are to be inserted in the plug-in unit, r=3.
n 3! 3! 3!
P r = = =6