Chapters 14 and 15Probability Basics
Probability FundamentalsCounting Rules Applied to the
Equally Likely Model
Birthday Problem• What is the smallest number of
people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?
• Answer: 23No. of people 23 30 40 60Probability .507.706.891.994
Probability
•Formal study of uncertainty•The engine that drives Statistics
• Primary objective of Chapters 14 and 15:
1. use the rules of probability to calculate appropriate measures of uncertainty.
2. Learn the probability basics so that we can do Statistical Inference
Introduction• Your favorite basketball team has the ball and trails by 2 points with
little time remaining in the game. Should your team attempt a game-tying 2-pointer or go for a buzzer-beating 3-pointer to win the game? (This situation has often been used in Microsoft job interviews).
• After a touchdown should a coach kick the extra point or go for two?
• On 4th down should your favorite football team punt or try for the first down?
• With a man on first base and no one out, should the manager call for a sacrifice bunt?
• If your favorite basketball team has a 3 point lead with little time left on the clock and the other team has the ball, should your team foul?
A phenomenon is random if individual
outcomes are uncertain, but there is
nonetheless a regular distribution of
outcomes in a large number of repetitions.
Randomness and probabilityRandomness ≠ chaos
Coin toss The result of any single coin toss is
random. But the result over many tosses
is predictable, as long as the trials are
independent (i.e., the outcome of a new
coin flip is not influenced by the result of
the previous flip).
The result of any single coin toss is
random. But the result over many tosses
is predictable, as long as the trials are
independent (i.e., the outcome of a new
coin flip is not influenced by the result of
the previous flip).
First series of tossesSecond series
The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.
The Laws of Probability
1. Relative frequencyevent probability = x/n, where x=# of occurrences of event of interest, n=total # of observations– Coin, die tossing; nuclear power plants?
• Limitationsrepeated observations not practical
Approaches to Probability
Approaches to Probability (cont.)
2. Subjective probabilityindividual assigns prob. based on personal experience, anecdotal evidence, etc.
3. Classical approachevery possible outcome has equal probability (more later)
Basic Definitions
• Experiment: act or process that leads to a single outcome that cannot be predicted with certainty
• Examples:1. Toss a coin2. Draw 1 card from a standard deck of
cards3. Arrival time of flight from Atlanta to
RDU
Basic Definitions (cont.)
• Sample space: all possible outcomes of an experiment. Denoted by S
• Event: any subset of the sample space S;typically denoted A, B, C, etc.Null event: the empty set Certain event: S
Examples1. Toss a coin once
S = {H, T}; A = {H}, B = {T}2. Toss a die once; count dots on upper
faceS = {1, 2, 3, 4, 5, 6}A=even # of dots on upper face={2, 4, 6}B=3 or fewer dots on upper face={1, 2, 3}
3.Select 1 card from adeck of 52 cards.S = {all 52 cards}
Coin Toss Example: S = {Head, Tail}Probability of heads = 0.5Probability of tails = 0.5
3) The complement of any event A is the event that A does not occur, written as A.
The complement rule states that the probability
of an event not occurring is 1 minus the
probability that is does occur.
P(not A) = P(A) = 1 − P(A)
Tail = not Tail = Head
P(Tail ) = 1 − P(Tail) = 0.5
Probability rules (cont’d)
Venn diagram:
Sample space made up of an event
A and its complement A , i.e.,
everything that is not A.
Birthday Problem• What is the smallest number of
people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?
• Answer: 23No. of people 23 30 40 60Probability .507.706.891.994
Example: Birthday Problem
• A={at least 2 people in the group have a common birthday}
• A’ = {no one has common birthday}
502.498.1)'(1)(
498.365
343
365
363
365
364)'(
:23365
363
365
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APAPso
AP
people
APpeople
Mutually Exclusive (Disjoint) Events
• Mutually exclusive ordisjoint events-no outcomesfrom S in common
A and B disjoint: A B=
A and B not disjoint
A
A
Venn Diagrams
Laws of Probability (cont.)
General Addition Rule
5. For any two events A and B
P(A or B) = P(A) + P(B) – P(A and B)
20
For any two events A and B
P(A or B) = P(A) + P(B) - P(A and B)P(A or B) = P(A) + P(B) - P(A and B)
A
B
P(A) =6/13
P(B) =5/13
P(A and B) =3/13
A or B
+_
P(A or B) = 8/13
General Addition Rule
Laws of Probability (cont.)
Multiplication Rule
6. For two independent events A and B
P(A and B) = P(A B) = P(A) × P(B)P(A and B) = P(A B) = P(A) × P(B)
Laws of Probability: Summary
• 1. 0 P(A) 1 for any event A• 2. P() = 0, P(S) = 1• 3. P(A’) = 1 – P(A)• 4. If A and B are disjoint events, then
P(A or B) = P(A) + P(B)• 5. For any two events A and B,
P(A or B) = P(A) + P(B) – P(A and B)• 6. for two independent events A and
B,P(A and B) = P(A) × P(B)
M&M candies
Color Brown Red Yellow Green Orange Blue
Probability 0.3 0.2 0.2 0.1 0.1 ?
If you draw an M&M candy at random from a bag, the candy will have one
of six colors. The probability of drawing each color depends on the proportions
manufactured, as described here:
What is the probability that an M&M chosen at random is blue?
What is the probability that a random M&M is any of red, yellow, or orange?
S = {brown, red, yellow, green, orange, blue}
P(S) = P(brown) + P(red) + P(yellow) + P(green) + P(orange) + P(blue) = 1 P(blue) = 1 – [P(brown) + P(red) + P(yellow) + P(green) + P(orange)]
= 1 – [0.3 + 0.2 + 0.2 + 0.1 + 0.1] = 0.1
P(red or yellow or orange) = P(red) + P(yellow) + P(orange)
= 0.2 + 0.2 + 0.1 = 0.5
Example: college students
• L = {student lives on campus}
• M = {student purchases a meal plan}
P(a student either lives or eats on campus)
= P(L or M) = P(L) + P(M) - P(L and M)
=0.56 + 0.62 – 0.42
= 0.76
Suppose 56% of all students live on campus, 62% of all students purchase a campus meal plan and 42% do both.Question: what is the probability that a randomly selected student either lives OR eats on campus.
Assigning Probabilities
If an experiment has N outcomes, then each outcome has probability 1/N of occurring
If an event A1 has n1 outcomes, then
P(A1) = n1/N
DiceYou toss two dice. What is the probability of the outcomes summing to 5?
There are 36 possible outcomes in S, all equally likely (given fair dice).
Thus, the probability of any one of them is 1/36.
P(the roll of two dice sums to 5) =
P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 / 36 = 0.111
This is S:
{(1,1), (1,2), (1,3), ……etc.}
Counting in “Either-Or” Situations• NCAA Basketball Tournament, 68
teams: how many ways can the “bracket” be filled out?
1. How many games?2. 2 choices for each game3. Number of ways to fill out the bracket:
267 = 1.5 × 1020
• Earth pop. about 6 billion; everyone fills out 100 million different brackets
• Chances of getting all games correct is about 1 in 1,000
Counting Example
In the knock-out stages of a soccer tournament, when a game ends in a tie the winner is determined by a penalty-kick shootout. The shootout, which consists of an alternating sequence of penalty kicks, is won by the team with the greatest goal tally after 5 kicks per team.
A coach has selected the 5 players that will take the penalty kicks in a shootout. In how many ways can the coach designate the order in which the 5 players take the penalty kicks?
Solution
There are 5 players to choose to take the first penalty kick, 4 remaining players to take the second penalty kick, 3 players for the third penalty kick, 2 players for the fourth penalty kick, and 1 player for the fifth penalty kick.
The number of possible arrangements is therefore
5 4 3 2 1 = 120
Efficient Methods for Counting Outcomes
Factorial Notation:n!=12 … n
Examples1!=1; 2!=12=2; 3!= 123=6; 4!
=24;5!=120;Special definition: 0!=1
Factorials with calculators and Excel
Calculator: non-graphing: x ! (second function)graphing: bottom p. 9 T I Calculator Commands(math button)
Excel:Insert function: Math and Trig category, FACT function
Permutations
A B C D EHow many ways can we choose 2
letters from the above 5, without replacement, when the order in which we choose the letters is important?
5 4 = 20
Permutations with calculator and Excel
Calculatornon-graphing: nPr
Graphingp. 9 of T I Calculator Commands(math button)
ExcelInsert function: Statistical, Permut
Combinations
A B C D EHow many ways can we choose 2
letters from the above 5, without replacement, when the order in which we choose the letters is not important?
5 4 = 20 when order importantDivide by 2: (5 4)/2 = 10 ways
ST 311 Powerball Lottery
From the numbers 1 through 20,choose 6 different numbers.
Write them on a piece of paper.
Chances of Winning?
760,38!6)!620(
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important.not order t,replacemen
without 20, from numbers 6 Choose
620206
C
Example: Illinois State Lottery
balls) pong pingmillion 16.5 house, ft (1200
months) 10in second 1about (
165,827,25!6!48
!54
importantnot order t;replacemen
withoutnumbers 54 from numbers 6 Choose
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654 C
North Carolina Powerball Lottery
Prior to Jan. 1, 2009 After Jan. 1, 2009
:
55!3,478,761
5!50!
:
42!42
1!41!
3,478,761*42
146,107,962
5 from 1- 55
1 from 1- 42 (p'ball #)
:
59!5,006,386
5!54!
:
39!39
1!38!
5,006,386*39
195,249,054
5 from 1- 59
1 from 1- 39 (p'ball #)
The Forrest Gump Visualization of Your Lottery Chances
How large is 195,249,054?$1 bill and $100 bill both 6” in length
10,560 bills = 1 mileLet’s start with 195,249,053 $1 bills
and one $100 bill …… and take a long walk, putting
down bills end-to-end as we go
Chances of Winning NC Powerball Lottery?
Remember: one of the bills you put down is a $100 bill; all others are $1 bills.
Put on a blindfold and begin walking along the trail of bills.
Your chance of winning the lottery is the same as your chance of selecting the $100 bill if you stop at a random location along the trail and pick up a bill .