10A-Probability


 * Probability Revision **


 * Discrete Random Variables **


 * ** Outcomes ** are the possible results of an experiment.


 * ** Sample Space ** (or Universal Set) is the list of all possible outcomes.


 * ** Discrete ** means the outcomes within the sample space are seperate and countable (as opposed to continuous).


 * ** Continuous ** means the outcomes can be any Real values between the endpoints - usually resulting from measuring (as opposed to counting).


 * An ** event ** is the set of one or more outcomes being described.


 * ** Probability ** is the likelihood (chance) of something happening. It is usually expressed as a fraction or decimal between 0 (impossible) and 1 (certain). It is sometimes expressed as a percentage between 0% and 100%.


 * Sets and Venn Diagrams **


 * A ** set ** is a collection of elements (or things, people, activities, numbers, ideas, etc)


 * ** Sets ** are often named with capital letters: A, B, C, etc


 * ** n(A) ** represents the __number__ of elements in the set called A.

A ** Venn Diagram ** is a way of representing one or more sets within the Universal Set.


 * A rectangle is used to denote the entire sample space (or the Universal Set)


 * A circle inside the rectangle denotes a set (a subset of the Universal Set)


 * The circle may contain
 * the elements within that set, or
 * the number of elements within the set.


 * Example **


 * ** E ** = {1, 2, 3, 4, 5, 6, 7, 8}


 * A = {1, 3, 5, 7}


 * B = {1, 2, 3, 4}


 * Union (OR) **


 * The Union of two events is A È B


 * It includes all outcomes in __**either**__ A **__or__** B


 * A È B = {1, 2, 3, 4, 5 7}


 * We sometimes use the word "**or**" instead of Union


 * Intersection (AND) **


 * The Intersection of two events is A Ç B


 * It includes only those outcomes which are in **__both__** A __**and**__ B


 * A Ç B = {1, 3}


 * We sometimes use the word "**and**" instead of Intersection


 * Complement (NOT) **


 * The Compl__e__ment of A is A'
 * not to be confused with "compl__i__ment" which is a nice comment about someone


 * It includes all outcomes that are **__not__** in A (or outside of A)


 * A' = {2, 4, 6, 8}


 * We sometimes use the word "**not**" instead of complement

NOTE:


 * Probability of an Event occurring = Pr(A) **

If each outcome is equally likely then: math . \qquad Pr(A) = \dfrac{ \text{Number of outcomes in A}}{\text{Total number of outcomes}} = \dfrac{n(A)}{n(\varepsilon)} \qquad. math

math . \qquad 0 \leqslant Pr(A) \leqslant 1 \qquad. math

... ... Pr(A) = 0 ... --> ... Event is impossible

... ... Pr(A) = 1 ... --> ... Event is certain


 * The sum of probabilities of all possible outcomes is always 1 **


 * Pr(A) + Pr(A') = 1
 * An event and its complement covers all possible outcomes


 * Pr(A) + Pr(B) + Pr(C) = 1
 * If A, B, C don't overlap and A or B or C covers all possible outcomes
 * If A, B, C don't overlap and A or B or C covers all possible outcomes


 * New notation: **

Sometimes we use p(x) instead of Pr(A) where p(x) represent the probability of event x happening.

math . \qquad \Sigma p(x) = 1 \qquad. math

A bag contains 5 black marbles, 7 red marbles (total = 12)
 * Example **

a) What is the probability that a randomly drawn marble is black?

Let B = Black Marble math . \qquad Pr(B) = \dfrac{5}{12} \qquad. math

b) What is the probability that a randomly drawn marble is not black?

math \\ . \qquad Pr(B') = 1 - Pr(B) \qquad. \\ . \\ . \qquad Pr(B') = 1 - \dfrac{5}{12} \\. \\ . \qquad Pr(B') = \dfrac{7}{12} math

The Addition Rule of Probability ... ... ** Pr(A È B) = Pr(A) + Pr(B) – Pr(A Ç B) **


 * The probability of __**either**__ A __**or**__ B happening is equal to:
 * The probability of A plus the probability of B minus the probability of both A __**and**__ B.
 * We have to subtract Pr(A Ç B) because otherwise we would be counting it twice.

Mutually Exclusive Events

Two events are ** mutually exclusive ** if they cannot occur at the same time

... ... A Ç B = {} so ... ... Pr(A Ç B) = 0

Therefore: ... .... Pr(A È B) = Pr(A) + Pr(B)

Independent Events

Two events are ** independent ** if the outcome of one does __not__ influence the outcome of the other.

Two events are ** dependent ** if the outcome of one __does__ influence the outcome of the other.

... .... Tossing one coin and then a second coin are ** independent **.
 * Example 1: **


 * Example 2: **
 * Drawing one card from a deck of playing cards and then drawing a second card without replacing the first one are two ** dependent ** events
 * because the probabilities for the second card are different depending on which card was drawn first.

If A and B are __**independent**__ then ... .... ** Pr(A Ç B) = Pr(A) x Pr(B) **

Also If .... ... Pr(A ** Ç ** B) = Pr(A) x Pr(B) then ... ... the events A and B are __independent__.


 * Example 3 **


 * An eight-sided dice has possible outcomes {1, 2, 3, 4, 5, 6, 7, 8}
 * If A = {1, 2} and B = {2, 3, 4, 5} show that A and B are independent.

__**Solution:**__

math \\ . \qquad Pr(A) = \dfrac{2}{8} = \dfrac{1}{4} \\. \\ . \qquad Pr(B) = \dfrac{4}{8} = \dfrac{1}{2} \\. \\ . \qquad Pr(A) \times Pr(B) = \dfrac{1}{4} \times \dfrac{1}{2} = \dfrac{1}{8} \qquad. math and math \\ . \qquad A \cap B = \{ 2 \} \\. \\ . \qquad Pr(A \cap B) = \dfrac{1}{8} \qquad. math

Since Pr(A Ç B) = Pr(A) x Pr(B) we have that A and B are independent.


 * Karnaugh Maps and Probability Tables **

These are both laid out the same except that
 * a ** Karnaugh Map ** shows the number of possible outcomes in each cell
 * a ** Probability Table ** shows the probability of that cell.


 * Example **

A bag of 20 beads contains 8 red beads and 12 blue beads. Draw a Karnaugh Map and a Probability Table for this situation.
 * Six of the red beads are cubes, the rest are spheres.
 * Six of the blue beads are cubes, the rest are spheres.


 * Solution:**


 * If A = red then A' = blue
 * If B = cube then B' = sphere




 * Conditional Probability **


 * Conditional probability ** is the probability of one event, given that another event has already happened.

The probability of A, given that B has happened is:

math . \qquad Pr(A \, | \, B) = \dfrac{Pr(A \cap B)}{Pr(B)} \qquad. math or math . \qquad Pr(A \cap B) = Pr(A \, | \, B) \times Pr(B) \qquad. math

Notice that ... ... Pr (A | B) is not equal to Pr (B | A)


 * Tree Diagrams **

Useful for both dependent and independent events

The probability of each outcome goes on the middle of each branch

The probability of a result is the __**product**__ of the probabilities on the branches leading to that result.

If more than one result meets the criteria for an event then the probability of that event is the __**sum**__ of the probabilities of the relevant results.


 * Example **

Nadine's car starts 70% of the time. If her car starts, she has an 80% chance of arriving at work on time. If her car doesn't start, she has a 50% chance of arriving at work on time. Find the probability that Nadine arrives on time.


 * Solution:**
 * Let C = Car Starts (so C' = car doesn't start)
 * Let O = On time (so O' = not on time)

... ... Pr(O) = Pr(CO) + Pr(C'O) ... ... Pr(O) = 0.56 + 0.15 ... ... Pr(O) = 0.71


 * Combinations **


 * Combinations ** are the number of ways that items can be selected from a set where the __**order**__ they are selected in is __**not**__ important.

For example, to select 3 letters out of {a, b, c, d} where the order is not important, there are 4 possible combinations. {abc, abd, acd, bcd}

The number of Combinations of n objects, selected r at a time, is given by math . \qquad ^nC_r = \left( \begin{matrix} n \\ r \\ \end{matrix} \right) = \dfrac{n!}{(n-r)! \,r!} \qquad. math

Recall that n! = n(n - 1)(n - 2) ... 3 × 2 × 1 where n can be any Natural Number (positive integer) Eg: 5! = 5 × 4 × 3 × 2 × 1 = 120


 * NOTE:** To make the Combinations rule work correctly, we define 0! = 1

The Classpad calculator can find both n! and nCr math . \qquad \textit{Enter: } \; \textbf{nCr} \, (4, 3) \; \textit{ to get } \; ^4C_3 \qquad. math
 * {In the Virtual Keyboard, mth Tab, CALC} **


 * Example **

A committee of 6 people is to be selected from a group of 10. How many combinations are possible?

math . \qquad ^{10}C_6 = \left( \begin{matrix} 10 \\ 6 \\ \end{matrix} \right) = \dfrac{10!}{4! \, 6!} = 210 \qquad. math

or use the calculator by entering **nCr(10, 6)** .