Chapter 5 Stats
What are the two rules of probability?
1. The probability of any event (E), must be greater than or equal to 0 or less than or equal to 1. 0-1 2. The sum of probabilities must equal 1.
A certain probability
100%
How many of each suit is in a deck?
13 diamonds 13 Spades 13 clubs 13 hearts
How many distinguishable strings of letters can be formed by using all the letters in the word REARRANGE?
15,120
The fixed-price dinner at a certain restaurant provides the following choices: Appetizer: soup or salad Entrée: baked chicken, broiled beef patty, baby beef liver, or roast beef au jus Dessert: ice cream or cheesecake How many different meals can be ordered?
2 x 4 x 2= 16 combos
How many cards are red and black in a deck?
26 red, 26 black
How many faces are in a deck of cards?
4 jacks 4 queens 4 kings (12 faces)
As a rule of thumb, if the sample size is less than ____% of the population size, we treat the events as independent.
5%
How many cards are in a deck?
52 cards
In 2005, 19.1% of all murder victims were between the ages of 20 and 24 years old. Also in 2005, 16.6% of all murder victims were 20-24 year old males. What is the probability that a randomly selected murder victim in 2005 was male given that the victim is 20-24 years old?
86.9%
The probability that a randomly selected female aged 60 years old will survive the year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that four randomly selected 60 year old females will survive the year?
96.78%
Sample Space
All possible outcomes (S). Ex: heads and tails
What is a permutation?
An arrangement of things in a particular order ie: how many different ways can you arrange 5 statues on a shelf? 5x4x3x2x1 = 120 different ways
What is an event?
Any possibility of outcomes from a probability experiment.
What is the law of large numbers?
As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.
Probabilities involving the phrase "at least" typically use the...
Complement rule
Suppose that of 100 circuits sent to a manufacturing plant, 5 are defective. The plant manager receiving the circuits randomly selects 2 and tests them. If both circuits work, the shipment is accepted. Otherwise, the shipment is rejected. What is the probability that the plant manager discovers at least 1 defective circuit and rejects the shipment?
.09798
Factorial, find the values: 0! 1! 2! 3! 4! 5! 6! 7! 8!
0!= 1 1!= 1 2!= 2 3!= 6 4!= 24 5!= 120 6!= 720 7!= 5,040 8!= 40,320
An impossible probability
0%
Complement of an event
E^C. All outcomes in a sample space that are not outcomes in event.
empirical probability distribution
Estimate based on relative frequencies of the observations.
What is an unusual event in probability?
Event with a low probability of occurring. Less than 5% chance
Multiplication Rule for Independent Events formula
If P and F are independent events, then P(E and F)=P(E)xP(F)
Multiplication rule of counting
If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, then the task of making these selections can be done in pxqxr different ways
General Multiplication Rule
In words, the probability of E and F is the probability of event E occurring times the probability of event F occurring, given the occurrence of event E.
P(E)=?
N(E)/N(S)
Suppose that a single six-sided die is rolled. What is the probability that the die comes up 4? Now suppose that the die is rolled a second time, but we are told the outcome will be an even number. What is the probability that the die comes up 4?
P(4)= 1/6 P(4, even #)= 1/3 P(4, 1,2,3,4)=1/4
If E and F are independent events, then...
P(E and F)= P(E)xP(F)
Classical Probability formula
P(E)= Number of ways that E can occur/ number of possible outcomes
Empirical probability formula
P(E)= Relative frequency of E= Frequency of E/Number of trials in experiment
If events E1, E2, E3,... En are independent, then...
P(E1 and E2 and E3 and... En)=P(E1)xP(E2)...P(En)
If events E1, E2, E3,.. En are independent, then...
P(E1 and E2 and E3... En)= P(E1)xP(E2)....xP(En)
Complement formula
P(E^C)= 1-P(E)
According to the American Veterinary Medical Association 31.6% of American households own a dog. What is the probability that a random ly selected household does not own a dog?
P(no dog)= 1- P(dog)= .684
What is an experiment?
Process that can be repeated in which the results are uncertain. Ex: Flipping a coin, picking a card
What is probability?
The measure of likelihood of something happening.
Two events E and F are dependent if...
The occurrence of event E in a probability experiment affects the probability of event F
conditional probability
The probability of event F given event E, P(EIF)
Conditional probability Rule
The probability of event F occurring, given the occurrence of event E, is found by dividing the probability of E and F by the probability of E, or by dividing the number of outcomes in E and F by the number of outcomes in E.
What are disjoint events?
Two events, no outcomes in common. Mutually exclusive. P(E or F)= P(E)+P(F)
A combination is...
a collection, without regard to order, in which r objects are chosen from n distinct objects with n greater than or equal to r, and without repetition. nCr
A survey of 10,000 Americans found that 600 had type O -negative blood. (a)Suppose we randomly select 1 of the 10,000 Americans surveyed. What is the probability that he or she has type O-negative blood? (b)If two individuals from this group are randomly selected, what is the probability that both have type O-negative blood? (c)Compute the probability of randomly selecting two individuals from this group who have type O-negative blood, assuming independence.
a. .06 b. .00359 c. .0036
Determine whether the events are dependent or independent A. E: Speeding on the interstate. F: Being pulled over by a police officer. B. E: You gain weight. F: You eat fast food for dinner every night. C. E: You get a high score on a statistics exam. F: The Boston Red Sox win a baseball game. D. E: Your favorite color is blue. F: Your friend's favorite hobby is fishing.
a. Dependent b. Dependent c. Independent d. Independent
Suppose that Ralph gets a strike when bowling 30% of the time. (a) What is the probability that Ralph gets two strikes in a row? (b) What is the probability that Ralph gets a turkey (three strikes in a row)? (c) When events are independent, their complements are independent as well. Use this result to determine the probability that Ralph gets a turkey, but fails to get a clover (four strikes in a row).
a. P(strike and strike)= .3(.3)= .09 b. P(turkey)= .3x.3x.3= .027 c. P(s,s,s, not s)= (.3)^3 x .7= .0189
Probabilities involving the phrase "at least" typically use the..
complement rule
An experiment is said to have equally likely outcomes when...
each simple event has the same probability of occurring.
Two events are dependent if...
if the occurrence of event in a probability experiment affects the probability of event
Probability model
lists the possible outcomes of a probability experiment and each outcome's probability
Permutations with nondistinct items
n!/ n1!x n2!x n3!x........nk!
Combination formula
nCr = n!/r!(n-r)!
What is the permutation formula?
nPr = n!/(n-r)!
subjective probability
probability based on an individual's opinion of the likelihood that an event will occur, or that an event or relationship is due to more than chance. Ex= weathermen
Two events E and F are independent if
the occurrence of event in a probability experiment does not affect the probability of event .
Two events E and F are independent if
the occurrence of event E in a probability experiment does not affect the probability of event F .
Whats the difference between disjoint events vs independent events?
two events are disjoint if they have no outcomes in common, that is, if knowing that one of the events occurs, we know the other event did not occur. Independence means that one event occurring does not affect the probability of the other event occurring. Therefore, knowing two events are disjoint means that the events are not independent.
