Unit 5

Question 1

A quick example of an blank would be rolling a three using a singular six-sided, fair dice. There is only one way this event can happen; there is only one favorable outcome. However, when you roll a six-sided dice, there are actually six possible outcomes. So, the probability of rolling a three is 1/6.

Answer 1

event

Question 2

The set of all six possible outcomes is referred to as the blank. Sample space is similar in nature to the universal set; that is, the set of all possible elements. The favorable outcome(s) is/are referred to as the event.

Answer 2

sample space

Question 3

One of the primary applications of counting is to calculate blank of random events.

Answer 3

probabilities

Question 4

An experiment is a procedure that results in one out of a number of possible blank.

Answer 4

outcomes

Question 5

The set of all possible outcomes is called the blank of the experiment.

Answer 5

sample space

Question 6

A subset of the sample space is called an blank.

Answer 6

event

Question 7

Blank is concerned with experiments in which the sample space is a finite or countably infinite set.

Answer 7

Discrete probability

Question 8

A set is blank if there is a one-to-one correspondence between the elements of the set and the integers.

Answer 8

countably infinite

Question 9

A set that is not countably infinite is said to be blank.

Answer 9

uncountably infinite

Question 10

Examples of blank sets include the set of all binary strings (of any length), the set of ordered pairs of integers (Z × Z), the set of all rational numbers

Answer 10

countably infinite

Question 11

The set of real numbers is an example of an blank.

Answer 11

uncountably infinite set

Question 12

Continuous and discrete probability are very similar in nature. For the purposes of this course in the application to computer science, the main difference is that discrete probability assumes you are working with either blank or blank.

Answer 12

finite or countably infinite sets

Question 13

Blank is the set of all possible outcomes.

Answer 13

Sample space

Question 14

An blank is the set of all favorable outcomes and is a subset of the sample space.

Answer 14

event

Question 15

An blank refers to the inability to provide a 1-to- 1 correspondence to the elements in the set. For example, just looking at the set of (real) numbers between 0 and 1, those values are truly uncountable-you can always find a number between any two numbers in the set.

Answer 15

uncountable infinite set

Question 16

An blank-the type we will be studying in the course-would include the integers, for example. You can create a 1-to- 1 correspondence between the elements of the set and the counting numbers. Even though it is an infinite set of numbers, there are no numbers between each element; they are separate and discrete.

Answer 16

infinitely countable set

Question 17

blank, which is this sum of probabilities.

Answer 17

probability distribution

Question 18

If all the probabilities for the event subsets are equal, then the distribution is known as a blank.

Answer 18

uniform probability distribution

Question 19

he probability distribution in which every outcome has the same probability is called the blank.

Answer 19

uniform distribution

Question 20

A blank is just the sum of each event subset’s probability. The sum of the distribution should always be 1 if the experiment is fair.

Answer 20

probability distribution

Question 21

If the probability of each element is the same, then this is a blank.

Answer 21

uniform distribution

Question 22

Use this formula to calculate the probability of an event:

Answer 22

probability of an event = number of favorable outcomes / total number or possible outcomes

Question 23

Two events are blank if the two events are disjoint (i.e., the intersection of the two events is empty).

Answer 23

mutually exclusive

Question 24

The probability of two events happening as an OR-that is, either this event A happens OR this event B happens-is a union of the event sets. If the two sets do not overlap, then they are referred to as blank.

Answer 24

exclusive

Question 25

To calculate the probability of the union of two mutually exclusive events blank the probability of each event.

Answer 25

add

Question 26

The blank for counting (with two sets) is similar to the blank principle in probability. If the event sets overlap, then you must subtract the probability of their intersection. To calculate the probability of two events that overlap, add the probability of each event then subtract the probability of the overlapping events.

Answer 26

inclusion-exclusion principle

Question 27

Using the idea of the blank of a set is very useful in calculating the probability of cumbersome events.

Answer 27

complement

Question 28

Just like when counting using the complement of a set, you can calculate the probability of an event by subtracting the probability of the event not happening from the probability of the event certainly happening. Use the formula:

Answer 28

p(E) = 1 - p(-E) (complement of E)

Question 29

If the event F happens, the new probability of E is the blank of E given F, denoted by p(E|F).

Answer 29

conditional probability

Question 30

Conditional probability is based on two events, A and B, which are dependent. If event B occurs first, the new probability (sample space for event A) has now changed. The probability for event A given that B occurred first is notated by (A|B) and is calculated using this formula:

Answer 30

P(A|B) = p(A Intersect B) / p(B)

Question 31

Remember, just because you’re working with dependent events (or overlapping, intersecting sets), it does not mean all the concepts you have used for counting do not also apply for probability. For example, you can use the rules of complements of sets or events to help calculate the blank.

Answer 31

conditional probability

Question 32

Two events are blank if conditioning on one event does not change the probability of the other event. Here is a formal description in terms of probabilities

Answer 32

independent

Question 33

If X and Y are events in the same sample space, and X and Y are independent, then

Answer 33

p(X ∩ Y) = p(X) · p(Y)

Question 34

To calculate the probabilities of two independent events, blank the individual probabilities

Answer 34

multiply

Question 35

Blank, based on conditional probabilities, allows you to update the probability of hypotheses when provided evidence from observations.

Answer 35

Bayes’ theorem

Question 36

The formula for Bayes’ theorem states the relationship between p(H)-the probability of the hypothesis before the evidence, and P(H|E) the probability of the hypothesis after getting evidence is blank

Answer 36

P(H|E) = P(E|H) / P(E) * P(H)

Question 37

Bayes’ theorem is very commonly used in determining the likelihood of blank. In other words, it is used to determine how likely a test or algorithm will produce an incorrect result. This theorem is important in factory applications (determining if the items being produced are defective or not), in medical testing (determining the accuracy of diagnostic tests), and many, many other computer science applications.

Answer 37

“false-positives.”

Question 38

A blank X is a function from the sample space S of an experiment to the real numbers. X(S) denotes the range of the function X.

Answer 38

random variable

Question 39

The blank of a random variable is the set of all pairs (r, p(X = r)) such that r ∈ X(S).

Answer 39

distribution

Question 40

A blank is a rule of association between each outcome of an experiment and some value defined by the rule.

Answer 40

random variable

Question 41

There is a connection between random variable probability distribution and the blank

Answer 41

binomial theorem

Question 42

The probability of an event in a sample space is also the blank. Therefore, to calculate the probability distribution of a random variable, apply the same concepts as you did in the “Probability of an Event” lesson in Module 18.

Answer 42

probability of the associated random variable value

Question 43

To calculate blank, create a probability distribution and multiply each outcome value by its probability, then add the results. If there is a cost to “play,” then the net expected value can be calculated by subtracting the cost from the expected value.

Answer 43

expected value

Question 44

Recall from your previous lesson, a random variable has values assigned from the outcomes of an experiment. Calculating the expected value of random variables is the blank as described in the previous bullet-same principle, different name.

Answer 44

same