Statistics -Probability

Probability Rule 2

P(S) = 1; that is, the sum of the probabilities of all possible outcomes is 1.

OR

In probability, either one or the other or both.

S

the sample space

Addition Rule for Multiple Disjoint Events

P(A or B or C) = P(A) + P(B) + P(C)

Disjoint

Two events that cannot occur at the same time are

A quiz consists of 10 multiple-choice questions, each with four possible answers, only one of which is correct. A student who does not attend lectures on a regular basis has no clue what the answers are, and therefore uses an independent random guess to answer each of the 10 questions. What is the probability that the student gets at least one question right? Let's decide on some notations. Let L be the event that the student gets at least one of the questions right. We'll use R for getting a question right and W for getting it wrong (W is essentially "not R"). Since L is an event of the type "at least one of ... ," it is much easier to find P(not L) and then use the Complement Rule. P(L) = 1 - P(not L). Write in words what the event "not L" is, and then find P(not L) using the Multiplication Rule.

"Not L" is the event that the student gets none of the questions right or in other words, "not L" is the event that the students gets all the questions wrong. Let W1 represent that the student gets the first question wrong that the student, W2 represent that the student gets the second question wrong, and so on. P(not L)=P(getting all the questions wrong)= P(W1 and W2 and W2 and W3 and W5 and W6 and W7 and W8 and W9 and W10). Now, using the multiplication rule, P(not L)= .75 * .75 * .75 * .75 * .75 * .75 * .75 * .75 * .75 * .75 = 0.0563.

A fair coin is tossed 10 times. Which of the following two outcomes is more likely? (a) HHHHHHHHHH (b) HTTHHTHTTH

(a) and (b) are equally likely. The 10 tosses are independent, so we'll use the Multiplication Rule for Independent Events: P(HHHHHHHHHH) = P(H) * P(H) * ... *P(H) = 1/2 * 1/2 *... * 1/2 = (1/2)10 P(HTTHHTHTTH) = P(H) * P(T) * ... * P(H) = 1/2 * 1/2 *... * 1/2 = (1/2)10

Probability Rule 3

P(not A) = 1 - P(A); that is, the probability that an event does not occur is 1 minus the probability that it does occur.

Why did we leave out the case when the events are disjoint and independent?

A and B disjoint means that they cannot happen together. In other words, A and B disjoint implies that if event A occurs then B does not and vice versa. Well... if that's the case, knowing that event A has occurred dramatically changes the likelihood that event B occurs - that likelihood is 0. This implies that A and B are not independent.

Probability Rule 1 Example

As previously discussed, all human blood can be typed as O, A, B, or AB. In addition, the frequency of the occurrence of these blood types varies by ethnic and racial groups. According to Stanford University's Blood Center (bloodcenter.stanford.edu), these are the probabilities of human blood types in the United States (the probability for type A has been omitted on purpose): Motivating question for rule 2: A person in the United States is chosen at random. What is the probability of the person having blood type A? Answer: Our intuition tells us that since the four blood types O, A, B, and AB exhaust all the possibilities, their probabilities together must sum to 1, which is the probability of a "certain" event (a person has one of these four blood types for certain). Since the probabilities of O, B, and AB together sum to .44 + .1 + .04 = .58, the probability of type A must be the remaining .42 (1 - .58 = .42):

Two-way table of probabilities cont

Essentially, as long as we are given (or can calculate) one cell in each of the margins (the total row and column), and one of the four cells in the body of the table, we'll be able to complete the entire table. Visually, we need:

Probability Rule 1

For any event A, 0 ≤ P(A) ≤ 1 The probability of an event, which informs us of the likelihood of it occurring, can range anywhere from 0 (indicating that the event will never occur) to 1 (indicating that the event is certain). One practical use of this rule is that is can be used to identify any probability calculation that comes out to be more than 1 as wrong.

General Addition Rule

For any two events A and B, P(A or B) = P(A) + P(B) - P(A and B).

Two people are chosen simultaneously and at random. What is the probability that both have the same blood type?

For both to have the same blood type there are four possibilities. Both have blood type O or both have blood type A or both have blood type B or both have blood type AB. P(same blood type) = P([O1 and O2] or [A1 and A2] or [B1 and B2] or [AB1 and AB2]) Since our four possibilities of both people having the same blood type are disjoint , using our Addition Rule we can add their probabilities (i.e., replace every "or" with +). Also, within each of the four possibilities, we can use the Multiplication Rule and replace "and" with * (using the same independence argument as the first example on this page). Our answer is therefore,

On the "Information for the Patient" label of a certain antidepressant it is claimed that based on some clinical trials, when taking this medication, there is a 14% chance of experiencing sleeping problems, or insomnia (denote this event by I), there is a 26% chance of experiencing headaches (denote this event by H), and there is a 35% chance of experiencing at least one of these two side effects (denote this event by L). What is the probability that a patient taking this drug will not experience insomnia?

Good job! Indeed, using the complement rule, P(not I) = 1 - P(I) = 1 - 0.14 = 0.86.

On the "Information for the Patient" label of a certain antidepressant it is claimed that based on some clinical trials, when taking this medication, there is a 14% chance of experiencing sleeping problems, or insomnia (denote this event by I), there is a 26% chance of experiencing headaches (denote this event by H), and there is a 35% chance of experiencing at least one of these two side effects (denote this event by L). In this context, the complement of event L, "not L," is the event that:

Good job! The event L is that the patient experiences at least one of the side effects. This means that for event L to occur, one of three things needs to happen: either the patient experiences I, or the patient experiences H or the patient experiences both I and H. Your answer is correct because it describes the only way in which event L will not occur: the patient does not experience either one of the two side effects.

The Multiplication Rule for Independent Events

If A and B are two independent events, then P(A and B) = P(A) * P(B).

If A and B are disjoint, then P (A and B must be ?

If events A and B are disjoint, they can never happen together. In other words the event "A and B" can never occur, and thus P(A and B) = 0.

Suppose that Jim is applying to two colleges; college A, a state university, and college B, an "Ivy League" school. He figures the following: The probability that he will be admitted to college A is 0.7 (denoted as A). The probability that he will be admitted to college B is 0.25 (denoted as B). There is a 25% chance he is admitted to neither college A nor college B (denoted as N). In this context, the complement of event N, "not N", is the event that:

Jim is admitted to at least one college. Good job. Event N is that Jim is admitted to neither college A nor college B. The compliment of N (or not N), therefore, means that it is not the case that Jim is not admitted to either of the colleges. In other words, 'not N' means that Jim is admitted to college A or college B, or both college A and college B. This is the same as saying that Jim is admitted to at least one of the two colleges.

A study reported that 37% of credit card users pay their bills in full each month. Two credit card users are chosen at random. What is the probability that the first one pays his/her bills in full and the second does not pay his/her bill in full?

Let A be the event that the first person does pays his/her bill in full each month, and let B be the event that the second person pays his/her bill in full each month. We need to find P (not A and not B) Since the two card users are chosen at random from a large population (of all card users) events 'A' ,'not B' are independent, and using the multiplication rule for independent events: P(A and not B)=[since independent]=P(A)*P(not B)=0.37*0.63

Two credit card users are chosen at random. What is the probability that both users do not pay their bills in full each month?

Let A be the event that the first person does pays his/her bill in full each month, and let B be the event that the second person pays his/her bill in full each month. We need to find P (not A and not B) Since the two card users are chosen at random from a large population (of all card users) events 'not A' ,'not B' are independent, and using the multiplication rule for independent events: P (not A and not B)=[since independent]=P(not A)*P(not B)=0.63*0.63

More than two independent events naturally occur

Multiplication Rule can be extended to more than two independent events. So if A, B and C are three independent events, for example, then P(A and B and C) = P(A) * P(B) * P(C). These extensions are quite straightforward, as long as you remember that "or" requires us to add, while "and" requires us to multiply.

Probability Rule 4 B

P(A and B) = P(event A occurs and event B occurs) finding P(A and B), the probability that both events A and B occur

Probability Rule 4

P(A or B) = P(event A occurs or event B occurs or both occur) P(A or B) = P(A) + P(B). The Addition Rule for Disjoint Events When dealing with probabilities, the word "or" will always be associated with the operation of addition; hence the name of this rule, "The Addition Rule."

On the "Information for the Patient" label of a certain antidepressant it is claimed that based on some clinical trials, when taking this medication, there is a 14% chance of experiencing sleeping problems, or insomnia (denote this event by I), there is a 26% chance of experiencing headaches (denote this event by H), and there is a 35% chance of experiencing at least one of these two side effects (denote this event by L). The probability of "not L" is:

P(not L) = 1 - P(L) = 1 - 0..35 = 0.86.

Magnitude of P(A or B) and of P(A and B) relative to either one of the individual probabilities.

Since probabilities are never negative, the probability of one event or another is always at least as large as either of the individual probabilities. Since probabilities are never more than 1, the probability of one event and another generally involves multiplying numbers that are less than 1, therefore can never be more than either of the individual probabilities.

A 2011 poll by the Pew Research Center for People and the Press estimated that 62% of U.S. adults favor the death penalty for persons convicted of murder, 31% oppose it, with the remaining 7% undecided. What is the probability that two randomly chosen U.S. adults support the death penalty for persons convicted of murder?

Since the two people are chosen at random from a large population, we can assume independence and use the multiplication rule of for independent events. P(1st supports and 2nd supports)=[independence]= P(1st supports)*P(2nd supports)= 0.62 * 0.62 = 0.3844

Using the Complement Rule

Suppose you know that the probability of getting the flu this winter is 0.43. What is the probability that you will not get the flu? Let the event A be getting the flu this winter. We are given P(A)=0.43. The event not getting the flu is A′. Thus, P(A′)=1−P(A)=1−0.43=0.57.

Finding the Probability of "At least one of ..."

The Complement Rule, P(A) = 1 - P(not A), together with the Multiplication Rule, is extremely useful for finding the probability of events like "at least one of ..." in several repetitions of a random experiment.

The Complement of an Event

The complement A′ of the event A consists of all elements of the sample space that are not in A.

Three people are chosen at random from a large population. What is the probability that all three have blood type B?

The events B1, B2, and B3 are independent since the three people were choosen at random from a large population and therefore using the multiplication rule extended P(B1 and B2 and B3) = P(A) * P(B) * P(C)

Specific Events

The word "and" is associated in our minds with "adding more stuff." Therefore, some students incorrectly think that P(A and B) should be larger than either one of the individual probabilities, while it is actually smaller, since it is a more specific (restrictive) event.

General Events

The word "or" is associated in our minds with "having to choose between" or "losing something," and therefore some students incorrectly think that P(A or B) should be smaller than either one of the individual probabilities, while it is actually larger, since it is a more general event.

Three people are chosen at random. (Assume the choices are independent events). What is the probability that they all have the same blood type?

To get all three the same, either the first and the second and the third are type O, or the first and the second and the third are type A, or the first and the second and the third are type B, or the first and the second and the third are type AB. The probability is: (.44 * .44 * .44) + (.42 * .42 * .42) + (.10 * .10 * .10) + (.04 * .04 * .04) = .160336

Using the Complement Rule

Two coins are tossed simultaneously. Let the event A be observing at least one head. What is the complement of A, and how would you calculate the probability of A by using the Complement Rule? Since the sample space of event A={HT,TH,HH}, the complement of A will be all events in the sample space that are not in A. In other words, the complement will be all the events in the sample space that do not involve heads. That is, A′={TT}. The second part of the problem is to calculate the probability of A using the Complement Rule. Recall that P(A)=1−P(A′). This means that by calculating P(A′), we can easily calculate P(A) by subtracting P(A′) from 1. P(A′)=P(TT)=1/4 P(A)=1−P(A′) =1−1/4 =3/4

Independent Events

Two events A and B are said to be independent if the fact that one of the events has occurred does not affect the probability that the other event will occur.

Two suicide cases are selected at random. What is the probability that both suicides were committed by a person of the same gender?

Two events fit the description "both suicides same gender:" MM and FF. These are disjoint events, so we can add P(both males) + P(both females). = (0.8)(0.8) + (0.2)(0.2) = 0.68

Venn Diagram

a simple way to visualize events and the relationships between them using rectangles and circles.

Event

a statement about the nature of the outcome that we're actually going to get once the experiment is conducted. Events are denoted by capital letters (other than S, which is reserved for the sample space).

Random Experiment

an experiment that produces an outcome that cannot be predicted in advance

Problems with "or"

if you solve a problem that involves "or," and the resulting probability is smaller than either one of the individual probabilities, then you know you have made a mistake somewhere.

Sample Space

list of possible outcomes

Probability of an Event

we can find the probability of any event A by dividing the number of outcomes in A by the number of outcomes in S: