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Chapter 4Probability Theory
4.1 What is Probability?
Law of Large Numbers
Jacob Bernoulli: “For even the most stupid of men… is convinced that the more observations have been made, the less danger there is of wandering from one’s goal”
Law of Averages is what people say when they assume that eventually they will win the lottery. Law of averages compensates for loss.
Law of Averages does not exist.
Imagine..
You have a hankering for an egg and cheese on a roll (ketchup, salt, pepper..). It is the first day of open campus for seniors, so during your free period you get in the car and drive down to Neils. You get to the light at the end of Martinsville road and it is red. Are you anxious? Do you worry about getting back in time, on this your first day of open campus?
The next day you have the same hankering… and the light is red again – what are the odds???
The following day.. .you guessed it. Would you then decide to go to O’Bagel?Do you really think that the probability of hitting the
red light is 100%Probably not…
Probability
Probability is the long run relative frequency of an event. Randomness eventually settles to probability.
Lets say you keep track….
1 Red 100% (1 out of 1)
2 Green 50% (1 out of 2)
3 Green 33% (1 out of 3)
4 Red 50% (2 out of 4)
5 Red 60% (3 out of 5)
6 Green 50% (3 out of 6)
Day Light is… % of time it is red
What would the graph look like over time
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8 9 10 11 12
Series1
There is no stop light elf in there watching for your car… Therefore, if the light is going to be red a certain percentage of the time, over time you should see the prediction level out.
Predicting
Predicting particular results is difficult (call heads or tails on a coin toss, win vs. loss for a football pool)
Long run prediction is easier for certain events (in the long run, the coin should be heads roughly 50% of the time)
Each trial is an AttemptWhat happen is the OutcomeCombination of outcomes is called the Event
Imagine this:
The probability of winning Mega Millions = 1/175,711,536.
Imagine 175,711,536 quarters in a row.One is purple on the underside. You will win
the lottery if you pick up the quarter that is purple.
How long is the row of quarters?(5280 feet in a mile)That would get us to….Fresno CA, if we stopped in San Francisco
first… (as the crow flies)
Probabilityhttp://www.ncaa.org/research/prob_of_competing/
http://anthro.palomar.edu/mendel/mendel_2.htm
http://www.wunderground.com/ndfdimage/viewimage?type=pop12®ion=us
Probability – the numerical measure of the likelihood of an event.
0 ≤ P(A) ≤ 1
Probability (cont)
What does it mean if P(A) is close to 0?
What does it mean if P(A) is close to 1?
What does it mean if P(A) = 0? = 1?
Probability (cont)
What does it mean if P(A) is close to 0?
What does it mean if P(A) is close to 1?
What does it mean if P(A) = 0? = 1?
fP(A) Relative Frequency =
n
Probability (cont)
Lets try rolling dice. You keep track, and when we are all done we will put the results on a giant chart…
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10
11
6 7 8 9 10
11
12
Predicted Frequency of 2, 12? .028
Predicted Frequency 3, 11? .056
Predicted Frequency of 4, 10? .083
Predicted Frequency of 5, 9? .111
Predicted Frequency of 6, 8? .139
Predicted Frequency of 7? .167
Probability (cont)
Was your probability close to predicted?
Did it get better the more sums considered?
For Equally likely outcomes,
Probability (cont)
Was your probability close to predicted?
Did it get better the more sums considered?
For Equally likely outcomes, # of ways favorable to AP(A)
total # of outcomes
Probability (cont)Some Definitions
Statistical Experiment (Observation) – any random activity that results in a definite outcome.
Event – a collection of 1 or more outcomes of a statistical experiment
Simple event – one outcome of a statistical experiment
Sample Space – set of all Simple Events
Sum of Probability of all Simple events = 1
Probability (contr)
The complement of event A = AC = describes the event NOT occurring
ThereforeP(A) + P(AC) = 1
4.2 Probability Rules
Cards, Dice, etc
Dependent vs Independent
First, we need to define an independent vs a dependent event.
Rolling two dice Tossing two coins
Drawing two cards from a deck
Drawing three marbles from a bag
What is a distinguishing factor of these four things
Independent events have no effect on each other. That is, tossing one coin has no impact on what you might get when you toss the second.
Dependent events do. Draw a card from a deck. Can you draw this card again? Not without replacement.
Independent EventsLets look back at our dice chart.What is the probability of rolling a 6 and 1 (in that
order)? What about rolling a 1 and 6 (in that order)?What is the probability of getting two sixes (chart)?
What is the relationship between those numbers?
P(A and B) = P(A) P(B)
Note: if A and B and C, then P(A)P(B)P(C)
Order matters!
Dependent Events
Kind of changes, but looks the same. That is, the probability of the second event will be slightly altered assuming success on the first. The basic concept is the same
P(A and B) = P(A) P(B, given A occurs)
P(A and B) = P(B) P(A, given B occurs)
Dependent Events (cont)
Drawing cards from a deck, without replacement, is a Dependent event. Once you draw the Ace of Hearts, you can’t draw it again.
What is the probability in Texas Hold’Em of being dealt two aces?
What is the probability of being dealt two red aces?
Conditional Probability
If P(A and B) = P(B) P(A, given B), then
Conditional Probability (cont)
If P(A and B) = P(B) P(A, given B), then
P(A and B)
P(A,given B) P(A B)P(B)
That bar notation means probability of A, given B has occurred…
Probability of two events happening together
Back to the dice:What is the probability of getting a
total of 3? Look at your chart…How many ways are there to get a 3? How does this affect probability? Probability of A or B (1 then 2 or 2 then
1)It looks like we….
Probability of two events happening together
Add them..Yes, typically
P(A or B) = P(A) + P(B)
As long as the events are mutually exclusive. That is, if they cannot occur together.
Could one of the dice be 1 and 2 at the same time?? (P(A)+P(B)=0)
Mutually Exclusive Events
Imagine a deck of cards. What is the probability of drawing a diamond OR an ace?
P(diamond) + P(ace)But is there overlap?What if you draw the ace of diamonds?How many ace of diamonds are there?How to deal with this?
Mutually Exclusive Events
If events are mutually exclusive, then P(A or B) = P(A) + P(B)
If events are not mutually exclusive, then
P(A or B) = P(A) + P(B) – P(A and B)
Back to dice
What is the probability of rolling a sum greater than 7?
What is the probability of rolling a sum 7 or greater?
We can count on the chart, but how would it be written?
M&M’s
In 2001 the maker of M&Ms decided to add another color. They surveyed kids in nearly every country and asked them to vote among purple, pink and teal. The global winner was purple.
In the US and Japan the results were:
Purple Pink Teal
US 42% 19% 37%
Japan 16% 38% 36%
M&M’s (cont)
1. What is the probability that a Japanese M&M’s survey respondent selected at random preferred pink or teal?
2. If we pick two Japanese respondents, what is the probability that they both selected purple?
3. If we pick three, what is the probability that at least one preferred purple?
Suspicious driving
Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood test, and 22% both tests. What is the probability that a randomly selected DWI suspect is given
A) A test?B) A blood test or a breath test, but not bothC) Neither test?
Same situation…
Are a blood test and breath test mutually exclusive?
Are they independent? (Independent means P(B│A)= P(B)
Probability of B happening given A occurs is the same as P(B)
P(A and B)
P(A B) P(A,given B)P(B)
Same situation…
Are a blood test and breath test mutually exclusive?
Are they independent? (Independent means P(B│A)= P(B)
Probability of B happening given A occurs is the same as P(B)
P(A and B)
P(A B) P(A,given B)P(B)
4.3 Trees and Counting
Trees
Consider how many ways a team can win or lose in a season…
Or how many sequences you can get if you toss a coin 3 times.
Or how many ways you can ride 4 particular roller coasters at Great Adventure.
A tree diagram allows you to look at all possibilities.
Trees (cont)
Lets set up a tree for that last situation.
The choices are El Toro, Rolling Thunder, Superman the Ultimate Flight, and Kinda Ka.
Trees (cont)
By labeling each branch with an appropriate probability, you can use the tree diagram to compute probability of a particular outcome.
In the reading there will be an example that discusses pulling balls out of urns.
Write the probabilities as fractions on each “branch” and then use the concepts from last section to compute P(A and B)
Application
According to a study by the Harvard School of Public Health, 44% of college students engage in binge drinking, 37% drink moderately and 19% abstain entirely. Another study published in the American Journal of Health Behavior, finds that among binge drinkers aged 21 to 34, 17% have been involved in alcohol related automobile accidents while among non-bingers of the same age, only 9% have been involved in such accidents.
What is the probability that a randomly selected college student will be a binge drinker that has had an alcohol related car accident?
We could do this with conditional probability
(That is, finding the probability of selecting someone who is a binge drinker AND a driver with an alcohol related accident)
Lets look at it from a tree point of view – this is sometimes organizationally a good way to consider…
It also is a good way to solve a problem that asks more than one question…
Going backwards
What if you instead wanted to know if a student has an alcohol related accident, what is the probability that the student is also a binge drinker?
Remember
Going backwards
What if you instead wanted to know if a student has an alcohol related accident, what is the probability that the student is also a binge drinker?
Remember
P(A and B)
P(A B) P(A,given B)P(B)
Tree gives P(accident | binge) but we want P(binge |accident)
Using the above formula, P(binge |accident) = P(binge and accident) P(accident)=.075/.108 (remember the
tree?)=69%
P(A and B)
P(A B) P(A,given B)P(B)
Trees (cont)
Why does this work? The Fundamental Theorem of Counting saysIf there are m1 ways to do a first task, m2
ways to do a second task, m3 ways to do a third task…… mn ways to do the nth task,
then the total possible “patterns” or ways you could do all the tasks is
m1· m2· m3...mn
Permutations
Now is the time we can introduce a few new mathematical operators (that you should already know)
! is called the factorial symbol
n! = n(n-1)(n-2)(n-3)…..13! = 3(2)(1) = 65! = 5(4)(3)(2)(1)= 1200! = 1
Calculators use a special formula to compute factorials; this is a large number formula but as result your calculator will give you an answer for 1.5! which is false
Permutations (cont)
So what is a permutation?
A permutation of “n” elements taken “r” at a time is an ordered arrangement (without repetition) of r of the n elements and it is called nPr.
Permutations (cont)
So what is a permutation? A permutation of “n” elements taken “r” at a
time is an ordered arrangement (without repetition) of r of the n elements and it is called nPr.
The thing to remember is that ORDER MATTERS!!
n r
n!P
(n r)!
How to recognize a permutation problem
The wording will imply somehow that order matters.
In how many different ways can you ride 5 out of 11 of the max rated rides at Great Adventure?
“different ways” means order matters
Combinations
What if order doesn’t matters?
How many combinations of 5 of the 11 max rated rides at Great Adventure are there?
Groupings, in which order doesn’t matter, are called combinations.
Smaller or larger?
Combinations (cont)
It looks like a permutation formula but with one crucial difference.
A Combination n elements, r at a time, is equal to
Dividing by r! gets rid of overlap
n!
r!(n r)!
