Chapter 3

Question 1

What is the Conditional Probability Formula?

Answer 1

To find the conditional probability that event A occurs given that event B occurs, divide the probability that both A and B occur by the probability that B occurs; that is, P (A|B) = P(A∩B)/P(B) [We assume that P(B) does not equal 0]

Question 2

What is the Multiplicative Rule of Probability?

Answer 2

P (A∩B) = P(A)P(B|A) or, equivalently P (A∩B) = P (B) P (A|B). 
If multiples (i.e. probabability 2 specific workers selected out of 10): 
1st Worker: P (A) = P(1v1)+P(1v2)+P(1v3) = 1/10+1/10+1/10 = 3/10. 
2nd Worker: P(B|A)=P(1v2)+P(1v3)=1/9+1/9 = 2/9 (since one worker chosen before already) 
Therefore: P(A∩B) = P(A)P(B|A)=(3/10)(2/9)= 6/90 = 1/15. 

*Key words “both” and “and” occur -> imply intersection -> implies we need to multiply

Question 3

What are Independent Events?

Answer 3

Events A and B are independent events if the occurrence of B does not alter the probability that A has occurred; that is, events A and B are independent if P(A|B)=P(A). When events A and B are independent. It is also true that P(B|A)=P(B). Events that are not independent are said to be dependent.

*Mutually exclusive events are always dependent.

Question 4

What is the Probability of Intersection of Two Events?

Answer 4

If events A and B are independent, then the probability of the intersection of A and B equals the product of the probabilities of A and B; that is, P(A∩B) = P(A) P(B).

The converse is also true: If P(A∩B)=P(A)P(B), then events A and B are independent.

Question 5

What is the Multiplicative Counting Rule?

Answer 5

You have k sets of elements, nv1 in the first set, nv2 in the second set…, and nvk in the kth set. Suppose you wish to form a sample of k elements by taking one element from each of the k sets. Then the number of different samples that can be formed is the product

nv1,nv2,nv3…nvk

When to use? Intersection (A and B)

Question 6

What is the Permutations Counting Rule?

Answer 6

Given a single set of N different elements, you wish to select n elements from the N and arrange them within n positions. The number of different permutations of the N elements taken n at a time is denoted by PNvn and is equal to

PNvn = N(N-1)(N-2)* — * (N-n+1) = N!/(N-n)!
Where n! = n (n-1)(n-2)…(3)(2)(1) and is called n factorial (for example, 5! = 54321=120).

**The quantity of 0! is defined to be 1.

When to use?

Question 7

What is the Partitions Counting Rule?

Answer 7

Suppose you wish to partition a single set of N different elements into k sets, with the first set containing nv1 elements, the second containing nv2 elements, … and the kth set containing nvk elements. Then the number of differerent partitions is

N!/nv1!nv2!—nvk!, where nv1+nv2+nv3+…+nvk=N

When to use?

Question 8

What is the Combinations Counting Rule?

Answer 8

if you are drawing n elements from a set of N elements without regard to the order of the n elements, then the number of different results is (Nvn) = N!/n!(N-n)!

[Note: The Combinations Rule is a special case of the partitions rule when k=2.]

When to use?

Question 9

What is the Baye’s Rule?

Answer 9

Can be applied when an observed event A occurs with any one of several mutually exclusive and exhaustive events, Bv1, Bv2,…,Bk. The formula for finding the appropriate conditional probabilities is as given below:

Given k mutually exclusive and exhaustive events, Bv1, Bv2,…,Bk such that P(Bv1)+P(Bv2)+—-+P(Bk)=1, and given an observed event A, it follows that P(B∩A/P(A)

= P(B)P(A|B)/P(Bv1)P(A|Bv1)+P(Bv2)P(A|Bv2)+—+P(Bvk)P(A|Bvk)

When to use?

Question 10

How to Select Probability Rules Given Different Compound Events?

Answer 10

Union (A or B) —- Addition Rule

  - Mutually Exclusive P(AUB)- P(A)+P(B)
  - Not Mutually Exclusive P (AUB) = P(A) + P(B) - P(A∩B)

Intersection (A and B) —- Multiplication Rule

• Independent P(A∩B)=P(A) * P(B)
• Dependent P(A∩B)=P(A|B)P(B)=P(B|A)P(A)

Complementary (not A) —– Rule of Complements
P(Ac)=1-P(A)

Conditional (A given B) ---- Conditional Rule
        -Independent P(A|B)=P(A)
         P(B|A)=P(B)
        -Dependent P(A|B) = P(A∩B)/P(B)
          P(B|A) = P(A∩B/P(A)

Question 11

What is an experiment?

Answer 11

An act or process of observation that leads to a single outcome that cannot be predicted with certainty.

Question 12

What is a sample point?

Answer 12

The most basic outcome of an experiment.

Question 13

What is a sample space?

Answer 13

(of an experiment) is the collection of all its sample points.

Question 14

What are the Probability Rules for Sample Points?

Answer 14

Let p represent the probability of sample point i. Then

1. All sample point probabilities must lie between 0 and 1 (i.e., 0

Question 15

What is an event?

Answer 15

A specific collection of sample points.

Question 16

What is the Probability of an Event?

Answer 16

The probability of an event A is calculated by summing the probabilities of the sample points in the sample space for A.

Question 17

What are the Steps for Calculating Probabilities of Events?

Answer 17

1. Define the experiment; that is, describe the process used to make an observation and the type of observation that will be recorded.
2. List the sample points.
3. Assign probabilities to the sample points.
4. Determine the collection of sample points contained in the event of interest.
5. Sum the sample point probabilities to get the probability of the event.

Question 18

What is a union?

Answer 18

The union of two events A and B is the event that occurs if either A or B (or both) occurs on a single performance of the experiment. We denote the union of events A and B by the symbol AUB. AUB consists of all the sample points that belong to A or B or both.

Question 19

What is an intersection?

Answer 19

An intersection of two events A and B is the event that occurs if both A and B occur on a single performance of the experiment. We write A∩B consists of all the sample points belonging to both A and B.

Question 20

What is a complement?

Answer 20

The complement of an event A is the event A does not occur - that is, the event consisting of all sample points that are not in event A. We denote the complement of A by A^C.

Question 21

What is the Rule of Complements?

Answer 21

The sum of probabilities of complementary events equals 1; that is

P(A)+P(A^C)=1.

Question 22

What is the Additive Rule of Probability?

Answer 22

The probability of the union of events A and B is the sum of the probability of event A and the probability of event B, minus the probability of the intersection of events A and B; that is,

P(AUB) = P(A) + P(B) - P(A∩B)

Question 23

What are mutually exclusive events?

Answer 23

Events A and B are mutually exclusive if A∩B contains no sample points - that is, if A and B have no sample points in common. For mutually exclusive events,
P(A∩B) = 0.

Question 24

What is Probability of Two Mutually Exclusive Events?

Answer 24

If two events A and B are mutually exclusive, the probability of the union of A and B equals the sum of the probability of A and the probability of B; that is, P (AUB) = P(A) + P(B).