## 7.1: Sample Spaces and Events

First, here are some basic definitions.
 Definition Example An experiment is an occurrence whose result is uncertain. Throw a pair of dice and then add the numbers facing up. An outcome is some specific result of the experiment that we observe. Any number from 2 to 12; for example, the following picture represents the outcome 7:  The sample space for the experiment is the set of all possible outcomes. The set of all numbers from 2 to 12: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Dice Simulation Here is a simulation of the above experiment: To throw the dice ("perform the experiment") press the "Throw Dice" button. The outcome is the sum of the numbers facing up, and will show on the left.  Q In an experiment where a pair of distinguishable dice (one red, one green) is thrown and the number facing up on each die is noted, the sample space is:

{1, 2, 3, 4, 5, 6} {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,2) (2,3) (2,4) (2,5) (2,6) (3,3) (3,4) (3,5) (3,6) (4,4) (4,5) (4,6) (5,5) (5,6) (6,6)

Q In an experiment where a pair of indistinguishable dice is thrown and the number facing up on each die is noted, the sample space is:

{1, 2, 3, 4, 5, 6} {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,2) (2,3) (2,4) (2,5) (2,6) (3,3) (3,4) (3,5) (3,6) (4,4) (4,5) (4,6) (5,5) (5,6) (6,6)

Q A coin is tossed three times in succession, and the total number of times heads comes up is noted. The sample space is:

 {0, 1, 2, 3} {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} {HHH, HHT, HTH, HTT, THH, THT, TTH}

You should consult Section 7.1 in Finite Mathematics or Finite Mathematics and Applied Calculus for many additional examples of sample spaces.

 Event Given a sample space S, an event E is a subset of S. The outcomes in E are called the favorable outcomes. We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E, that is, if the outcome of the experiment is favorable. How To Determine The Set E Just say this to yourself when you are looking for the event E: The event E consists of all outcomes in S which are favorable. Example Imogen enjoys sitting in front of the TV and randomly grabbing two chocolates at a time from her snack jar. The snack jar contains a large number of nut log swirls, turkish delights, and mocha surprises. Describe a possible sample space, and also the event that Imogen selects at least one mocha surprise in her first grab. Solution Here, the elements of the sample space S can be taken to be combinations of two types of chocolates or pairs of the same type. So, a possible sample is the set of all such combinations: S = {NT, NM, TM, NN, TT, MM}, where N = nut log swirl, T = turkish delight, and M = mocha surprise. Now for the event E. Using the above suggestion, write down the following: The event E consists of all outcomes in S which are favorable. Since the favorable outcomes are those with at least one mocha surprise, we have The event E consists of all outcomes in S which contain at least one moca surprise. Thus, E = {NM, TM, MM}. (Just delete those outcomes containing no M.)

The next example is similar to Example 1 in Section 7.1 of Finite Mathematics and Finite Mathematics and Applied Calculus.

A U.S. factory worker in 2009 may or may not have been covered by medical insurance plan. If the worker was covered, the coverage could either have been under the employer's plan or another plan. In the case of another plan, the plan could either have been in the worker's name or in that of his or her spouse. Consider the experiment "Select a U.S. factory worker and determine whether the worker was covered and the type of coverage."

Q An appropriate sample space is:

 S = {Not covered, Covered} S = {Not covered, Covered by employer's plan, Covered by own plan, Covered by spouse's plan} S = {Not covered, Covered by employer's plan, Covered by own plan, Covered by spouse's plan, Covered by some other type of plan}

Q The event that a worker is not covered by the employer's plan is:

 E = {Not covered, Covered by own plan, Covered by spouse's plan} E = 3/4 E = 3 {Covered by own plan, Covered by spouse's plan}

Q A coin is tossed three times, and the sequence of heads and/or tails is noted. The event that heads comes up at least twice is:

 {2, 3} Two out of four {HHH, HHT, HTH, THH} {HHT, HHH}

Operations on Events

Since events are sets, we can ask ourselves what effects the set operations like union, intersection, and complement have on events:
Set Operation
Example
The complement, E', of an event E is the event that E does not occur. It is the set of all outcomes not in E. Toss a pair of dice and then add the numbers facing up. If E is the event that the sum is even, then E' is the event that the sum is odd:
 S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} E = {2, 4, 6, 8, 10, 12} E' = {3, 5, 7, 9, 11}
The union, E F, of events E and F is the event that either E occurs or F occurs (or both). Toss three coins and record the sequence of heads and tails. If E is the event that heads come up only once, and F is the event that tails come up only once, then E F is the event that either heads come up only once, or tails come up only once.
 S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} E = {HTT, THT, TTH}, F = { HHT, HTH, THH} E F = {HTT, THT, TTH, HHT, HTH, THH}

The intersection, E F, of events E and F is the event that both E and F occur. Pick a three-digit number (000-999) at random. If E is the event that the first digit is 9, and F is the event that the remaining digits add up to 2, then E F is the event that the first digit is 9 and the remaining digits add to 2.
 S = set of all three-digit numbers (1,000 of them)! E = the set of all numbers 900 through 999 F = the set of all numbers of the form *02, *11, or *20 E F = {902, 911, 920}

If E and F are events then E and F are said to be disjoint or mutually exclusive if E F is empty. In the experiment immediately above, take E to be the event that the first digit is 9 and F to be the event that the first digit is 8. Then E and F are mutually exclusive.

Q In the experiment where a pair of dice (one red, one green) is thrown and the number facing up on each die is noted, let E be the event that the sum of the numbers is 4, and let F be the event that the sum is an odd number. The event F' is:

 the event that the outcome is an even number {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6),(5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)} {2, 4, 6, 8, 10} the event that the outcome is an even number other than 4

Q With E and F as described above, E F' is the event

 {(1, 1), (1, 5), (2, 4), (2, 6),(3, 3), (3, 5), (4, 2), (4, 4),(4, 6), (5, 1), (5, 3), (5, 5),(6, 2), (6, 4), (6, 6)} {(1, 3), (2, 2), (3, 1)} same as F' the empty set

Q With E and F as described above, E' F is the event

 that the sum of the numbers is an odd number other than 4 that the sum of the numbers is any number that the sum of the numbers is any even number other than 4 that the sum of the numbers is any number other than 4

For more practice, try some of the questions in the chapter quiz (warning: it covers the whole of Chapter 7) by pressing the button on the sidebar. Then try the exercises in Section 7.1 of Finite Mathematics and Finite Mathematics and Applied Calculus

Last Updated: April, 2009
Copyright © 1999, 2003, 2009 Stefan Waner

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