Chapter 6: Probability
Section 6.2: Probability Models
A Little Probability Background
The mathematics of chance/randomness is called Probability.
Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. For example, toss a coin –in the long run, about ½ will be heads P(heads) = ½.
Samples are random – we are uncertain who will be chosen, but overall everyone has an equal chance of being selected.
Definition of Randomness and Probability. We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of
repetitions. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions
Examples. If we roll a 6 sided die, the probability of getting a 4 is 1/6. Flip a (fair) coin and the probability of “heads” is 1/2.
Note: Probabilities can be expressed as a fraction, decimal, or as a percent. The most common ways are as fractions or decimals.
Myths of Probabilities
Myth of short-run regularity – Remember that the patterns of probability apply to the long run. Myth of the law of averages -- Often random events are independent of previous events. This means that, knowing the outcome of one trial does not changed the probabilities for the outcomes of any other trials. For instance just because a fair coin has been flipped heads six times in a row, it is NOT more likely that the seventh flip will be tails.
Probability Models
The sample space S of a random phenomenon is the list of all possible outcomes.
An event is any collection of outcomes in the sample space, i.e. a subset of the sample space. Examples: Find the sample space
A probability model is a mathematical description of a random phenomenon consisting of two parts: A sample space S and a way of assigning probabilities to events.
What is the sample space in each of these situations? Toss a coin once
Toss a coin 2 times
Toss 3 coins
Roll 1 die
Roll 2 dice and sum of the up faces
Fundamental Counting Principle A.K.A Multiplication Principle
If you can do one task in a number of ways and a second task in b number of ways, then both tasks can be done in number of ways. For example. If you go to a pizzeria and they have 3 different types of crusts, (thin, Sicilian, stuffed) 7 different toppings (pepperoni, sausage, onions, BBQ chicken, bacon, salami, mushrooms) and 5 different beverages (water, soda, lemonade, HI-C, and beer) How many different combinations of crust, topping and beverage can you have?
Examples:
1.) The standard Connecticut license plate is three letters, followed by 3 numbers. How many license plates can you form using these specifications?
2.) How many 5 digit zip codes are possible?
Probability Rules
1. Any Probability is a number between 0 and 1. An event with probability 0 never occurs, and an event with probability 1 occurs on every trial.
In symbols:
2. The sum of the probabilities of all possible outcomes must equal 1.
In symbols: P(S) = 1
3. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.
In symbols: If A and B are “disjoint” a.k.a. “mutually exclusive” P(A or B) = P(A) + P(B) Note: This rule is also known as the addition rule for mutually exclusive events.
4. The probability that an event does not occur is 1 minus the probability that the event does occur.
In symbols: or
Note: This rule is also known as the complement rule.
Other notation:
In your AP Stat packet, they use set notation to represent the words “or” and “and” So P(A or B) = P(A) + P(B) can also be seen as P(A B) = P(A) + P(B)
In summary: “or” and are used interchangeably “and” and are used interchangeably
5. Two events A and B are independent if knowing that one occurs does not change the
probability that the other occurs. If A and B are independent then the probability of A and B is the product of their individual probabilities.
In symbols: If A and B are independent, P(A and B) = P(A) • P(B) Note: This rule is known as the multiplication rule for independent events. Other tidbits:
The multiplication rule applies only to independent events; you cannot use it if events are not independent.
Section 6.3: General Probability Rules
In this section we will consider some additional laws that govern assignment of probabilities. The purpose of learning more laws of probability is to be able to give probability models for more complex random phenomena.
Let’s take a closer look at the addition rule
3. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.
In symbols: If A and B are “disjoint” a.k.a. “mutually exclusive” P(A or B) = P(A) + P(B) Note: This rule is also known as the addition rule for mutually exclusive events.
Here is a Venn diagram for disjoint events
What happens if A and B are not disjoint,
This leads us to the General Addition Rule for Unions of Two Events For any two events A and B,
P(A or B) = P(A) + P(B) – P(A and B) S
A
B
S
A
P(A B) = P(A) + P(B) – P(A B)
Examples
1.) Zack has applied to both Princeton and Stanford. He thinks the probability that that Princeton will admit him is 0.4, the probability that Stanford will admit him is 0.5, and the probability that both will admit him is 0.2.
(a) Make a Venn diagram marked with the given probabilities.
(b) What is the probability that neither university admits Zack?
(c) What is the probability that he gets into Stanford but not Princeton?
2.) Suppose
55% of adults drink coffee 25% of adults drink tea 45% of adults drink cola 15% drink both coffee and tea 5% drink all three beverages 25% drink both coffee and cola 5% drink only tea
Draw a Venn diagram marked with the information. Use it along with the addition rules to answer the following questions
(a) What percent of adults drink only cola?
Conditional Probability
The probability we assign to an event can change if we know that some other even has occurred. This idea is the key to many applications of probability.
Notation: The probability of B given that A occurred is P(B│A) . Conditional probability is generally solved intuitively.
Examples:
Refer to the table to answer the following questions. 3200 people were surveyed.
Likes Chocolate Ice
Cream
Likes Vanilla
Ice Cream Strawberry IceLikes Cream
Total
Under 30 604 366 322
30-51 404 424 286
Over 51 336 72 386
Total 3200
a.) Find the probability that a randomly selected individual is over 51.
b.) Find the probability that a randomly selected individual likes vanilla ice cream.
c.) Find the probability that a randomly selected individual is under 30 or likes chocolate ice cream.
d.) Find the probability that a randomly selected individual is over 51 and likes vanilla ice cream?
e.) What is the probability you choose someone who likes strawberry ice cream given that you selected someone under 30?
f.) What is the probability you selected someone under 30 given that you selected someone who likes strawberry ice cream?
g.) Suppose you randomly choose an individual who is between 30 and 51, what is the probability the person you chose like chocolate ice cream?
Ice cream preference
Let’s take a closer look at the multiplication rule.
5. Two events A and B are independent if knowing that one occurs does not change the
probability that the other occurs. If A and B are independent then the probability of A and B is the product of their individual probabilities.
In symbols: If A and B are independent, P(A and B) = P(A) • P(B)
Note: This rule is known as the multiplication rule for independent events. Independence cannot be pictured by a Venn diagram, because it involves the probabilities of the events rather than just the outcomes that make up the events.
So what happens if the events are not independent (dependent)? This leads us to the General Multiplication Rule for Any Two Events The joint probability that events A and B both happen can be found by
P(A and B) = P(A) • P(B│A) Equivalently
P(A B) = P(A) • P(B│A)
Here P(B│A) is the conditional probability that B occurs, given the Information that A occurs.
Examples
1.) 29% of Internet users download music files, and 67% of downloaders say they don’t care if the music is copyrighted. Find the percent of internet users who download music and don’t care about copyright.
2.) In Texas hold ’em, you are dealt 2 cards. A great starting hand would be 2 aces. What is the probability you are dealt 2 aces?
Take a close look at the 2 formulas for the multiplication rule.
If events A and B are independent P(A B) = P(A) • P(B) If events A and B are dependent P(A B) = P(A) • P(B│A)
So how can you show (prove) that 2 events are independent?
Example: Using the ice cream and age 2-way table, are liking chocolate ice cream and being over 51 independent? Justify your answer.
Last thing:
If you take a look at general multiplication rule, you can algebraically manipulate it so that you get a formula for conditional probability:
P(A B) = P(A) • P(B│A) → P(B│A) =
Example:
