Math Math Mathy Math Flashcards
What is a permutation?
An order of arrangements of [r] objects without repetition, selected from [n] distinct objects. Where you need to count the # of ways you can arrange items where order is important. ORDER MATTERS!
Permutation formula
nPr = n!/(n-r)!
Permutations: 7 factorial
7!
Permutations: 0!
Always = 1
Permutations example: How many ways can 8 cd’s be arranged on a shelf?
[n] = 8 (8 cd's in this problem) [r] = 8 (arranging all 8 cd's) nPr = n!/(n-r)! nPr = 8!/(8-8)! nPr = 8!/0! = 8!/1 =8! =8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 =40320
Permutations example: In how many ways can a sorority of 20 members select a president, vp, and treasurer, assuming that the same person cannot hold more than one office?
[n] = 20 (20 members in the sorority) [r] = 3 (3 offices to be held) nPr = 20!/(20-3)! = 20!/17! = 6840
Permutations example: How many different arrangements can be made using 2 of the letters of the word TEXAS, if no letter is to be used twice?
[n] = 5 (5 letters in TEXAS) [r] = 2 (2 letters at a time) nPr = 5!/(5-2)! = 5!/3! = 20
What are combinations?
When you need to count the # of groupings, without regard to order.
Arrangement of [r] objects, without regard to order and without repetition, selected from [n] distinct objects.
Combinations formula
nCr = n!/(n-r)! r!
Combinations example: In a conference of 9 schools, how many intraconference football games are played during the season if the teams all play each other exactly once?
[n] = 9 (9 teams in conference) [r] = 2 (2 teams play per game) nCr = 9!/(9-2)! 2! = 9!/7! 2! = 36
Combinations example: You are going to draw 4 CARDS from a standard deck of 52 cards. How many different 4 card hands are possible?
[n] = 52 (52 cards in the deck) [r] = 4 (we want 4 hands) nCr - 52!/(52-4)! 4! = 52!/48! 4! = 270725
Combinations example: 3 marbles are drawn at random from a bag with 3 red and 5 white marbles. How many different draws are there?
[n] = 8 (8marbles in the bag) [r] = 3 (3marbles are drawn at a time) nCr = 8!/(8-3)! 3! = 8!/5! 3! = 56
What is probability?
Used to find out chances, research, and advertising
Probability: experiment definition
An act for which the outcome is uncertain (rolling a die, tossing a coin)
Probability: sample space definition
[S] set of all possible outcomes of the experiment such that each outcome corresponds to exactly one element in [S]. Elements of [S] are called sample points. If there is a finite # of sample points, that # is denoted [n(S)]
Probability: sample space example
Rolling a die would be: S = {1, 2, 3, 4, 5, 6}
Tossing a coin: S = {heads, tails}
Surveying favorite soft drinks: S = {sprite, coke, dr.p}
Probability: event definition
Subset [E] of a sample space experiment
Probability: event example
Rolling a single die: an event [E] could be rolling an even # is E = {2, 4, 6}
Tossing a coin: an event [E] could be tossing a tails is E = {tails}
Empirical Probability
Based on direct observations or experiences, represents the probability that an event E will occur
Empirical probability formula
P(E) = # of times x event occurs / total # of observed occurrences = probability that an event E will occur
Probability example: The table below lists the results of a student survey pertaining to favorite ethnic foods. Each student chose only one type of ethnic food for the survey. Italian: 15 Chinese: 20 Japanese: 3 Thai: 4 Mexican: 30 Other: 10 What is the probability that the student's favorite ethnic food chinese?
n(E) = n(students whose favorite ethnic food is chinese) n(S) = n(students surveyed)
n(E) = 20 (20 students liked chinese food best) n(S) = 82 (82 total students surveyed) = 20/82 = 10/41 = 2.44
Mean
The average
Median
The middle # after they have been put into numerical order
Mode
that occurs most often on the list
Range
Difference between largest and smallest #s
Standard Deviation formula
> Find the average of # set
Subtract each number in the set from the average
Square each of those numbers
Average the set of squared numbers (= variance)
Find square root of variance (= standard deviation)
