CS 260-01 Chapter 4 Notes

NAME	COUNTING	PROBABILITY	COMMENTS	SET THEORY
These notes are taken from several sections of charpter 4. They attempt to relate counting events with probability notation and, where possible, set notation. We have already seen a number of counting method in earlier chapters. [NOTE: p is probability. I will be adding examples and additional notes periodically over the next few days.
Sum Rule	n=n₁+ n₂	p(n₁+n₂)=p(n₁) + p(n₂)	total number of tasks when task 2 occurs after task 1 and both can't be done at the same time. There is no overlap. If we have 10 men and 7 women, we can chose from 17 people.	\|A U B\| = \|A\| + \|B\| A ^ B = Ø
Product Rule INDEPENDENT EVENTS (one outcome is not affected by another)	n=n₁* n₂	p(n₁n₂)= p(n₁) p(n₂)	the number of ways of doing task 1 and then doing task 2. If there are 26 ways to get the first letter of a password and 10 ways to get the second number, there are 26*10 ways to form a password with the first character A..Z and the second character as a digit.	\|A ^ B\| = \|A\|*\|B\|
Inclusion-Exclusion	n=n₁+ n₂- n₁* n₂	p(n₁+n₂)=p(n₁) + p(n₂) -p(n₁) * p(n₂)	total number of tasks when task 2 and task 1 could be done at the same time. There IS overlap. If there are 10 math majors and 15 computer science major and 5 people with math and computer sciences as major, we have 10+15-5 people.	\|A U B\| =\|A\| + \|B\| - \|A^B\|
Conditional Probability		P(E)*P(F) P(E\|F) ----------- P(F) if E and F are independent events	The probability that event E occurs given that event F HAS occured. If we want to know the probability of a head a from we role 5 on a dice, we have (1/6)*(1/2) ------------ (1/6)	p(E^F) -------- p(F)
r-permutations	P(n,r)=n(n-1)(n-2)... (n-r+1)		the number of ways of arranging n objects given r slots where ORDER MATTERS and REPITITION IS NOT ALLOWED Chose a President, Vice Presedent and Secretary from 6 people. P(6,3)=654 The first position can be filled 6 ways; after that, the next position can be filled 5 ways, after that, the next position can be filled 4 ways.
r-combinations	n! C(n,r) = -------- r!(n-r)! n things takes r at a time. P(n,r) ------- P(r,r)		the number of ways of arranging n objects given r slots where ORDER DOES NOT MATTER and RETETITION IS NOT ALLOWED. Chose a 3 co-captians from 6 people. C(6,3)=65454321 ------------------- 321321 We can select 3 people form 6 in order, but then there are 3! arrangements of this order. P(n,r)/P(r,r)=C(n,r)
permutation with replacements	n^r		If we can pick two letters from 26 and can choose the same letters, we have 26*26 choices,
combinations when repition is allowed	C(n+r-1,r)		A backery has 4 cookie types. You want to select 6 cookies and you can take more than on of each kind. C(4+6-1,6)=C(9,6)=84
permutation of indistinguishable objects	n! ------------ n₁! n₂! n₃!...		If you want to know the number of 4 combinations you can make with the word MISSISSIPPI, S occurs 4 times, P occurs 2 times I occurs 3 times M occurs once. Since there are 11 letters, we have 11! -------------- 4! 2! 3! 1! This is an extention of the combination formula.

Example Using Shifting of Expectation Values

Find the mean of 20,000, 30,000, 40,000

20000=aX+b
30000=aX+b
40000=aX+b

Let a=10000, b=0 then
X=2,3, 4
E(X)=3
a*E(X)+b= 10000*3+0=30000

Find the mean of 6.32, 6.34 and 6.36.

6.32=aX+b
6.34=aX+b
6.36=aX+b

Let a=1, b=6 then
X=.32, .34, .36
E(X)=(.32+.34+.36)/3 = .34
a*E(X)+b = 1* (.34) +6 = 6.34

This shows that we can find the mean of numbers when the decimal point is shifted. This is done very often without thinking about it.

This is used less often, but shows that adding a constant to numbers changes the mean by that constant.

Expectation Notation
E(X)	mean	E(X)=sum of the p(x)*value of X	This is the the sum of products of the probability of an event times the value of its outcome. See the example of grades that we did in class.
VAR(X)	variance	VAR(X) = the square of the difference between the observations and the meantimes the probability of the observations.	This is the square of the average distance between the observation and the mean. The square root is the average distance between the observation and the mean and is called the standard deviation. The standard deviation tells us how far from the mean an observation is. observations are within 1 standard deviation 68% of the time.

An example of Calculation Variance and Expectaion
Let X= 0,1, 2 heads on a two coin toss: p(X=0)=1/4, p(X=1)=1/2 and p(X=2)=1/4 Then the E(X)= 01/4 + 11/4 + 2* 1/4 = 1 VAR(X)= 1/4(0-1)² + 1/4(1-1)² + 1/4(2-1)² = 1/4 + 1/4 =1/2	p is the probability that X= the given number of heads. The probability of 0 heads is 1/4, 1 head is 1/2 and 2 heads is 1/4. The expectation is the probability of the event times the value of the outcome. The variance is sum of the squares of the difference between the outcome and the expectation times the probability of that outcome..

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