Probability


 

A phenomenon is random, if individual outcomes are uncertain, but the longterm pattern of many individual outcomes is predictable.
The probability of an outcome is the proportion of times the outcome would occur in a very long series of repetitions.
Sample space S = set of all possible outcomes for an experiment
an event = subset of the sample space S.

a probability model = a sample space S and a way of assigning probabilities to events.
example: for rolling a fair dice the sample space S = {1, 2, 3, 4, 5, 6}and the probability model is P(1) = P(2) = ...P(6) = 1/6.
A = {2, 3, 4, 5} is an event with P(A) = 2/3.

The complement of an event A is the event, that A does not occur, Ac .
example: A = event that the sun shines today, so Ac = the event that the sun does not shine today.

Two events are disjoint events, if they have no outcomes in common, also called mutually exclusive events.
example: In a car sales lot with 100 cars, 15 cars are red and 22 are silver. The events A = {red cars} and B = {silver cars} are disjoint.
P(A) = .15, P(B) = .22.

Two events are independent events if the occurrenceof one event has no effect on the other event.
example: A woman gives birth to her first child, it's a boy. Then she gives birth to a second child, it's a girl. These events are independent. The fact that she gave birth to a boy the first time has no influence over the fact that the second child was a girl.

A discrete probability model is one with a countable sample space.

examples:

First digits of numbers on legitimate records (checks, tax returns, invoices etc) follow Benford's law; the outcomes are not equally likely.

 
First digit 
1
2
3
4
5
6
7
8
9
Probability    0.301   0.176   0.125   0.097   0.079   0.067   0.058   0.051   0.046

What is the probability that the first digit is greater than 6?
If outcomes are equally likely, then the probability of event A, p(A) = (number of outcomes in A) / (number of outcomes in S)

problem 1:    Fair Dice

a) Throw two fair dice, a green one and a blue one and record the eyes thrown on the green one and the eyes thrown on the blue one.

b) Throw two fair dice, a green one and a blue one and record the sum of the eyes thrown. c)  What is the probability
blue/green
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

Probability Rules:

    The probability P(A) of any event A satisfies ): 0 ?= P(A) ?= 1.

    If S is the sample space in a probability model, then P(S) = 1.

    The Complement Rule:  P(Ac) = 1 - P(A).

    The Multiplication Rule for Independent events:  P(A and B) = P(A) x P(B).

    The general Addition Rule: P(A or B) = P(A) + P(B) - P(A and B).

examples:
    P(throwing  6 on a fair dice) = 1/6.

    P({!, 2, 3, 4, 5, 6}) = 1.

    P(not {4, 5} on a fair dice} = 1 - 2/6 = 4/6 = 2/3

    P(throw a 4, and then throw another 4 on a fair dice) = 1/6 * 1/6.

problem 2: What is the probability of problem 3: On a car sales lot there are 100 cars.
85 cars have CD player,
93 have air conditioning.
At leaast how many cars have both air conditioning and a CD player?

problem 4: On a car sales lot there are 100 cars.

85 cars have CD player,
93 cars have air conditioning.
83 cars have both AC and CD player
What is the probability

 
 

What is the probability of at least one occurrence of A in n independent trials?
 

 

p(at least one A in n independent trials) = 1 – (p(not A))n
 

problem 5:  What is the probability of getting at least 1 head when tossing 3 fair coins?
problem 6:  What is the probability of getting rain at least once in 10 days if the probability of rain each day is 0.1?

problem 6b:  What is the probability of getting at least one five hundred year flood in thirty years?
Apropos Harvey


 

Combinatorics is the study of counting

A permutation is an ordered arrangement of k items that are chosen without replacement from a collection of n items.
       P(n,k) = n*(n-1)*(n-2)* ...*(n-k+1)

a combination is an unordered arrangementof k items that are chosen without replacement from a collection of n items.
       C(n,k) =n*(n-1)*(n-2)* ...*(n-k+1) / k! = n!/[k!(n-k)!]

     
Choosing k items from n distinct items
 
repetition allowed repetition not allowed
order matters n*n*n*...*n = nk n*(n-1)*(n-2)* ...*(n-k+1)
order doesn't matter (n+k-1)!/[k!(n-1)!] n!/[k!(n-k)!]
problem 7: from a baseball team of 25 players, the manager chooses a lineup of 9 hitters.

problem 8: A club with 20 members needs to hold an election for president, secretary and treasurer. In how many ways can these positions be filled, if a member can only hold one position.

problem 9: In the Texas Hold 'Em style of poker, play begins with each poker player being dealt two cards face down. From a standard 52-card deck, how many possible 2-card hands could be dealt to you?

problem 10: In poker, a royal flush is a 5-card hand containing (in any order) an Ace, King, Queen, Jack, 10 all of the same suit.


 

A density curve is a curve that

problem 11: Generate two random numbers between 0 and 1 and take their sum. The sum can take any value between 0 and 2. The density curve
is the shaded triangle.
Note: the areas under a normal curve are probabilities.


 

The Mean (Expectation)

given sample space S with outcomes x1, x2..., xn with  respective probabilities p1, p2 ..., pn
then the mean is:
                m = x1 p1 + x2 p2 + ... + xn pn

the variance Var is:
                  Var  =  (x1 - m)2p1 + (x2 - m)2p2 + ... +( xn- m)2pn
the standard deviation s  is the square root of the variance.

The mean of a continuous probability model is the point at which the density curve would balance.
 

problem 12: Lottery probabilities:

Prize Probability
Jackpot 1 in 80,000,000
$100,000 1 in 2,000, 000
$5,000 1 in 400,000
$100 1 in 9,000
$100 1 in 8,000
$7 1 in 200
$7 1 in 700
$4 1 in 200
$3 1 in 70
            Find the expected value (mean) for a single lottery ticket. Then state how much you you can expect to win/lose if you spend $365 per year.

Problem 13:
A roulette wheel has 38 slots  numbered 00, 0, 1, 2, ..., 36, The ball is equally likely to land in any one of them when the wheel is spun. The slot numbers are laid out on a board on which gamblers place their bets. Say you place a bet on all the red numbers. There are 18 red numbers and 18 black. The remaining two  (0 and 00) are green. Betting costs 1$, and pays out 2$, if you win.

Fill in the table of outcomes for your bet:
xi pi
win
don't win

 

Law of Large Numbers

when a random phenomenon is repeated a large number of times


Gambler's fallacy : mistaken belief that a streak of bad luck makes a person "due" for a streak of good luck.

        Fill in the table:

 
 
n even mean of all rolls sofar standard deviation  $ win/loss 
first 100 45 .45     -10
next 100 47   (45+47)/200 = .46     -16
next 300 148   (92+148)/500 = .48     -20

       a)    How many even numbers would you have to roll in the next 100 rolls to break even? Is this likely?
       b)    Explain, how this illustrates the law of large numbers, even while your losses increased.


 

Central Limit Theorem

Draw a simple random sample (SRS) x of size n from any large population with mean m and standard deviation s, then

Problem 14: The scores of eighth-grade students on the National Assessment of Educational Progress (NAEP) year 2007 math test have a distribution that is approximately normal, with mean  m = 281 and standard deviation s = 35.