7.1: Sample Spaces and Events: Game Version
First, here are some basic definitions.
|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:
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.
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
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:
Since the favorable outcomes are those with at least one mocha surprise, we have
(Just delete those outcomes containing no M.)
Which two of the following are possible sample spaces? (You must click on the two correct answers consecutively, otherwise you will lose points and health!)
The text for this question will not appear until you have answered the preceding question correctly.
Operations on Events
Since events are sets, we can ask ourselves what effects the set operations like union, intersection, and complement have on events:
|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:
|The union, EF, 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 EF is the event that either heads come up only once, or tails come up only once.
|The intersection, EF, 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 EF is the event that the first digit is 9 and the remaining digits add to 2.
|If E and F are events then E and F are said to be disjoint or mutually exclusive if EF 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.|
Consider an experiment where a pair of dice (one red, one green) is thrown and the number facing up on each die is noted. Let A be the event that . The event A' is the event that:
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