Basics of Probability

Trial or Experiment Experiment - a process that results in a particular outcome or “event”. Simple event (or sample point), E i – an event that can’t be decomposed into multiple individual outcomes. Sample space, S - The set of all possible sample points for the experiment. Event, A i - a subset of the sample space.

Likelihood of an Outcome define the "likelihood" of a particular outcome or “event” where an event is simply a subset of the sample space. Assuming each sample point is equally likely,

A Simple Experiment jar contains 3 quarters, 2 dimes, 1 nickels, and 4 pennies, consider randomly drawing one coin. The sample space: Let A be the event that a quarter is selected

Drawing a Quarter? Randomly draw a coin from the jar... There are 3 quarters among the 10 coins: Assuming each coin is equally likely to be drawn.

Roll the Dice Using the elements of the sample space: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Considering the sum of the values rolled,

Roll the Dice Using the elements of the sample space: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Count the members for this event.

Roll the Dice Using the elements of the sample space: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Likelihood one of the faces shows a “2” ? P( a “2” is rolled ) =

2). Summing over all the sample points in the sample space… “a 100% chance one of the outcomes occurs” and… Properties of a Probability Each outcome A i in the sample space is assigned a probability value P( A i ) 1). “between a 0% and a 100% chance of occurring”:

Properties of a Probability …and 3). When a set of events A 1, A 2, A 3,… is pairwise mutually exclusive… P( “2” is rolled OR sum is greater than 8 ) =

The Sample-Point Method Define the sample space: describe and list the simple events, being careful not to include any compound events. Assign a probability to each sample point, satisfying the “properties of a probability”. Define the event of interest, A, as a set of sample points. Compute P(A) by summing the probabilities of sample points in A.

Sticky Spinner Suppose a game uses a spinner to determine the number of places you may move your playing piece. Suppose the spinner tends to stop on “3” and “6” twice as often as it stops on the other numbers. What is the probability of moving a total of 9 spaces on your next 2 spins?

Multiplication Principle ( called “mn rule” in text )

Cross Product and Power Set By the multiplication principle, if | A | = m and | B | = n, then | A x B | = mn. By the multiplication principle, if | A | = n, then

“Decision Tree” Total of 24 different systems Computer 3 choices Scanner 4 choices Printer 2 choices

Addition Principle For any two sets A and B, In particular, if A and B are disjoint sets, then

Extended to 3 sets… May generalize further for any n sets.

So for probability… Leads to an “addition rule for probability”:

Additive Rule of Probability and if events A and B are mutually exclusive events, this simplifies to

Either Way Note we can do addition first, then convert to a probability ratio: Or we can construct the probabilities, then do addition:

Compute the Probability (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Compute the Probability Given the following probability table: sedan mini-van truck totals male female If one of the owners is randomly selected…

Counting Permutations The number of ways to choose and arrange any r objects chosen from a set of n available objects, when repetitions are not allowed. “Permutations of n objects, taken r at a time”

Gold, Silver, Bronze Consider the top 3 winners in a race with 8 contestants. How many results are possible? Or equivalently,

Calculate it Calculators have a built-in feature for these computations (labeled as nPr ). Use the MATH button and PRB submenu. To compute the value we simply enter: 8 nPr 3

Compare the 2 cases Case 1: If a president, VP, and treasurer are elected, how many outcomes are possible? (select and arrange 3, order is important) 16 x 15 x 14 = 3360 pres. VP treas. Consider a club with 16 members: Case 2: If a group of 3 members is chosen, how many groups are possible ? (a choice of 3 members, order is not important) Since we don't count the different arrangements, this total should be less.

Adjust the total Case 1: Case 2: Given one group of 3 members, such as Joe, Bob, and Sue, 6 arrangements are possible: ( Joe, Bob, Sue), ( Joe, Sue, Bob), ( Bob, Joe, Sue) ( Bob, Sue, Joe), ( Sue, Joe, Bob), ( Sue, Bob, Joe) Each group gets counted 6 times for permutations. Divide by 6 to “remove this redundancy”.

Counting Combinations “Combinations of n objects, taken r at a time” when repetitions are not allowed Often read as “n, choose r" Sometimes denoted as

All Spades? For example, in a 5-card hand, P( all 5-cards drawn are spades)

4 Spades, and a Non-Spade? For example, in a 5-card hand, P(exactly 4 spades in 5-card hand)?

All Possible Cases? Consider the possible number of spades: P(all 5 spades) = P(exactly 4 spades) = P(exactly 3 spades) = P(exactly 2 spades) = P(exactly 1 spade) = P(no spades) =

Exactly 3 Face Cards? “3 face cards” implies other 2 cards are not face cards P( 5-card hand with exactly 3 face cards) = ?

Probable Committee? If a 3-person committee is selected at random from a group of 6 juniors and 9 seniors, what is the probability that exactly 2 seniors are selected? Setup the ratio, this type of committee as compared to all possible 3-person committees.

Binomial Coefficients Recall the Binomial Theorem: For every non-negative integer n… Remember “Pascal’s Triangle”?

Multinomial Coefficients “Ways to partition n objects into k groups” when repetitions are not allowed Here the groups are of size n 1, n 2, …, and n k such that n 1 + n 2 + … + n k = n.

Expanding a Multinomial Using the multinomial coefficients Determine the coefficient of the x 3 y 3 z 2 term. In the expansion of the multinomial