Section 4.1 Experiments, Sample Spaces, and Events

Experiments An experiment is an activity with speciﬁed observable results.

Outcomes An outcome is a possible result of an experiment.

Before discussing sample spaces and events, we need to discuss sets, elements, and subsets. Sets A set is a well-deﬁned collection of objects.

Elements The objects inside of a set are called elements of the set. a ∈ A ( a is an element of A ) a∈ / A ( a is not an element of A)

Roster Notation for a Set Roster notation will be used exclusively in this class, and consists of listing the elements of a set in between curly braces.

Subset If every element of a set A is also an element of a set B, then we say that A is a subset of B and we write A ⊆ B.

Sample Spaces and Events

A sample space, S, is the set consisting of ALL possible outcomes of an experiment. Therefore, the outcomes of an experiment are the elements of the set S, the sample space.

An event, E, is a subset of the sample space. The subsets with just one outcome/element of the sample space are called simple events. The event ∅ is called the empty set, and represents the event where nothing happens.

1. An experiment consists of tossing a coin and observing the side that lands up and then rolling a fair 4-sided die and observing the number rolled. Let H and T represent heads and tails respectively.

(a) Describe the sample space S corresponding to this experiment.

(b) What is the event E1 that an even number is rolled? (c) What is the event E2 that a head is tossed or a 3 is rolled?

(d) What is the event E3 that a tail is tossed and an odd number is rolled?

2. The numbers 3, 4, 5, and 7 are written on separate pieces of paper and put into a hat. Two pieces of paper are drawn at the same time and the product of the numbers is recorded. Find the sample space.

Set Operations

Set Union Let A and B be sets. The union of A and B, written A ∪ B, is the set of all elements that belong to either A or B or both. This is like adding the two sets. Below is a Venn Diagram illustrating the set A ∪ B.

A ∪ B

A B

2 Fall 2018, © Maya Johnson Set Intersection Let A and B be sets. The intersection of A and B, written A ∩ B, is the set of all elements that belong to both A and B. This is what the two sets have in common. Below is a venn diagram illustrating the set A ∩ B.

A ∩ B

A B

Complement of a Set If S is the sample space of a given experiment, and A is an event of S, then the set of all outcomes in S that are not in A is called the complement of A and is denoted Ac. Below are venn diagrams illustrating the events Ac and Bc.

Ac Bc

3. Let S = {d, g, k, q, v, y} be a sample space of an experiment and let E = {d, g} and F = {d, q, y} be events of this experiment. Find the events E ∪ F , E ∩ F , Ec, Ec ∩ F , E ∪ F c, and (E ∩ F )c.

3 Fall 2018, © Maya Johnson Example 3 continued...

Number of Outcomes in an Event E If E is an event of a sample space, then n(E) is the number of outcomes in the event E.

Number of Events of a Sample Space Suppose S is the sample space of a given experiment, and that n(S) = m, the number of simple events is m, where m is any nonnegative integer. Then the total number of events of S is 2m.

4. Let S = {5, 9, 12} be a sample space associated with an experiment.

(a) How many simple events does S have? How many events does S have in total?

(b) List all events of this experiment.

If E and F are events of a sample space S, then

n(E ∪ F ) = n(E) + n(F ) − n(E ∩ F )

Language: The word “and” means intersection (∩), while the word “or” means union (∪).

5. A experiment consists of asking 100 people at a particular cafe what type of drink they ordered and observing the responses. If there were 50 people who responded that they ordered coffee, 60 who responded that they ordered tea, and 20 who responded that they ordered coffee and tea, how many responded that they ordered coffee or tea?

5 Fall 2018, © Maya Johnson 6. A jar contains 8 marbles numbered 1 through 8. An experiment consists of randomly selecting a marble from the jar, observing the number drawn, and then randomly selecting a card from a standard deck and observing the suit of the card (hearts, diamonds, clubs, or spades).

(a) How many outcomes are in the sample space for this experiment?

(b) How many outcomes are in the event “a number more than 1 is drawn and a red card is drawn?”

6 Fall 2018, © Maya Johnson 7. Two fair 6-sided dice are rolled and the numbers shown uppermost are observed. Find the number of outcomes in the following events.

(a) The sum of the numbers is 7.

(b) A 5 is rolled.