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Wize University Statistics Textbook > Probability

Introduction to Probability

Measure of VariabilityPrevious Section
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Events & Sample Spaces


Experiment - process where you observe or measure the outcome, and the outcome cannot be predicted with certainty - it happens by chance

Trial - one run of the experiment

Event - a subset (combination) of outcomes that you are interested in (e.g. Event A, Event B...)

Sample space - a set of descriptions (S) that cover all possible outcomes of the experiment

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Sample Space Examples

  • Flipping a coin S = {Heads, Tails}
  • Rolling a dice and flipping a coin S = {(1&H), (2&H), (3&H), (4&H), (5&H), (6&H), (1&T), (2&T), (3&T), (4&T), (5&T), (6&T)}
  • Drawing a ball from a bag with 1 Red and 1 Green ball (repeating with replacement 3 times) S = {RRR, RRG, RGR, GRR, RGG, GRG, GGR, GGG} →\rightarrow these are all the possible outcomes

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Types of Probability

The probability of an event E is the chance or likelihood that the event E will happen is denoted by P(E)P\left(E\right).

Theoretical Probability

It is calculated by dividing the number of desired outcomes by the total number of all possible outcomes that could occur in theory.

The theoretical probaiblity that event E occurs is
P(E)=Number of ways E can happenTotal number of possible outcomes\displaystyle\boxed{P\left(E\right)=\frac{\text{Number of ways E can happen}}{\text{Total number of possible outcomes}}}

Example
  • You flip an ordinary coin twice.
  • The sample space (a set showing all possible outcomes) is {HH, HT, TH, TT}\left\{HH,\ HT,\ TH,\ TT\right\}.
  • The probability that you get tails twice is 14\frac{1}{4}

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Relative Probability

It is calculated by dividing the number of desired outcomes by the number of trials in an actual experiment.

The relative probability that event E occurs is
P(E)=Number of times we observed ENumber of trials \displaystyle\boxed{P(E)=\displaystyle \frac{\text{Number of times we observed E}}{\text{Number of trials }}}

Example
  • We toss an ordinary coin 200 times (# trials) and get heads 60 times (# observations).
  • Based on this experiment, we conclude that the probability of heads is 60200=0.30\frac{60}{200}=0.30
  • Since the probabilities of 'Heads' and 'Tails', in this experiment, are not equally likely (i.e. the events have equal probability), then we consider this an unfair or biased coin.
In the long run with infinitely many trials, the relative probability will approach closer and closer to the theoretical probability (actual probability). This is called the Law of Large Numbers.

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Subjective Probability

It is the probability that’s guessed by an individual based on their knowledge.

Example
  • You believe that you will ace your exam after this session with a 70% probability.
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Probability Language & Notation

Union

The union of 2 events A and B includes all the outcomes in A and B. This is denoted by A∪BA\cup B.
Example

AA is the event of rolling a number lower than 4 on a die.
A={1,2,3}A=\{1,2,3\}
BB is the event of rolling an odd number on a die.'
B={1,3,5}B=\{1,3,5\}

A∪B=A\cup B= {Event A or Event B}
A∪B={1,2,3,5}A\cup B=\{1,2,3,5\}

Notice that we listed '1' and '3' only once each.

Wize Tip
The keyword associated with unions is "or".


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Intersection

The intersection of 2 events A and B includes all the outcomes that are overlapping in both A and B. This is denoted by A∩BA\cap B.

Example

AA is the event of rolling a number lower than 4 on a die.
A={1,2,3}A=\{1,2,3\}
BB is the event of rolling an odd number on a die.
B={1,3,5}B=\{1,3,5\}
A∩B=A\cap B= {Event A and Event B}
A∩B={1,3}A\cap B=\{1,3\}

Notice that only '1' and '3" overlap in both events.

Wize Tip
The keyword associated with intersections is "and"

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Complement

The complement of Event A includes all the outcomes that are not in Event A. This is denoted by A′A' or AcA^c or A‾\overline{A}.

Examples

AA is the event of rolling an even number on a die. A′A' is the event of not rolling an even number on a die

AA is the event that you pass your exam
A′A' is the event that that you do not pass your exam

AA the event that two people will get along
A′A' is the event that two people will not get along
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Example: Probability Language

Consider these sets:
A={1,2,3,4,5}B={1,2,6}C={3,5}A = \{1, 2, 3, 4, 5\} \\ B = \{1, 2, 6\} \\ C = \{3, 5\}



Find the following:

A∩B=A∩ B =
{1, 2}



B∩C=B ∩ C =
ϕ→ Mutually exclusive\phi\rightarrow\ Mutually \ exclusive

A∪B=A ∪ B =
{1, 2, 3, 4, 5, 6}


A‾=\overline{A}=
{6}
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Independent vs. Mutually Exclusive Events

Independent Events

Events A and B are independent if the occurrence of one does not affect the probability of the other to occur.

Example
Flipping "heads" from a coin does not affect the probability of rolling a "5" on a die; it's still P("5") = 16\frac{1}{6}
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Mutually Exclusive Events (a.k.a. Disjoint)

Events A and B are mutually exclusive if they cannot both occur at the same time.

Note: A∩B=∅A\cap B=\varnothing ( the empty set)

Example 1
A →\to John was a home with his wife at 5pm
B →\to John was at the bank at 5pm

It is impossible for A and B to occur at the same time →\to mutually exclusive


Example 2

A →\to Susan can't find her keys.
B →\to Susan spills her coffee.

It is possible for both A and B to occur →\to not mutually exclusive

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Example: Independent Events

There are 4 black and 3 white balls in an urn. Two balls are drawn, one at a time, without replacement.
  • Event A →\rightarrow First ball is black
  • Event B →\rightarrow Second ball is black
(a) Are the events A and B independent? Mutually Exclusive?

A and B affect each other so they are not independent.
They can both happen so not mutually exclusive.

(b) How do your answers change if we replace the balls each time?

By replacing them the events are independent because Event A will not affect Event B, but they still are not mutually exclusive because they can both still happen.

Independent vs. Mutually Exclusive Events


You have two fair dice: Die #1 and Die #2.

Let Event A = the sum of the two dice is an odd number
Let Event B = the dice are the same number

Which of the following is true? (Check all the apply.)