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**Math 409/409G History of Mathematics**

The Probability of Simple Events

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What is probability? Probability is the measure of the likelihood of a random phenomenon or chance behavior.

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Definitions A simple event is any single outcome from a probability experiment. A sample space, S, of a probability experiment is the collection of all simple events. An event is any collection of outcomes from a probability experiment. An event consist of one or more simple events.

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Example A probability experiment consists of rolling a single “fair” die. What are the simple events of this probability experiment? What is the sample space? Give two examples of events that are not simple events.

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Answers The simple events are the possible outcomes of rolling the die. Since there are 6 numbers on the die, the simple events are: “Rolling a 1” {1} “Rolling a 2” {2} “Rolling a 3” {3} “Rolling a 4” {4} “Rolling a 5” {5} “Rolling a 6” {6}

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**The sample space, S, is the set of all simple events. So **

An event consists of one or more simple events. So two non-simple events are: E {2, 4, 6} “Rolling an even number” O {1, 3, 5} “Rolling an odd number”

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Another Definition A probability experiment is said to have equally likely outcomes if each simple event in the sample space has the same chance (probability) of occurring. Example: When rolling a “fair” die, each number on the die has the same chance of occurring as any other number. But if you replace the number 6 on the die with the number 2, then you no longer have equally likely outcomes since rolling a 2 has a better chance or occurring.

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**Computing Probabilities**

If an experiment has n equally likely simple events and if the number of ways that event E can occur is m, then the probability of event E is

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Examples A probability experiment consists of rolling a single “fair” die. Then:

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Important Properties If S is the sample space and E is an event in that sample space, then:

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**Mutually Exclusive Events**

Two events are mutually exclusive if they cannot occur at the same time. Example: When rolling a single fair die, the events of rolling a 1 and of rolling a 2 are mutually exclusive events since the die will show only one number. But the events of rolling a 1 and of rolling an odd number are not mutually exclusive since 1 is an odd number.

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**Addition Rules If A and B are mutually exclusive events, then**

If A and B are not mutually exclusive events, then

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Examples When rolling a single fair die, what’s the probability of rolling a 1 or a 2?

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**When rolling a fair die, what is the probability of rolling a 1 or rolling an odd number?**

First note that event of rolling a 1 and rolling an odd number is the same as the event of rolling a 1.

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Independent Events Two events A and B are independent if knowing whether A occurs does not change the probability that B occurs. Example: Two marbles are drawn one at a time from an urn containing 3 red marbles and 2 blue marbles. Are the events of first drawing a red marble and then drawing another red marble independent events?

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**Well, that depends on whether or not the first marble is placed back in the urn.**

If the marble is put back in the urn after drawing the first marble, then the event of first drawing a red marble and then drawing another red marble are independent events since after putting the marble back, the sample space remains the same. So the probability of drawing the second red marble is the same as the probability of drawing the first red marble.

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But if the first marble is not put back, then after the first marble is drawn, the sample space has been reduced by one marble. So the probability of the second marble has changed after drawing the first marble. So the events of first drawing a red marble and then drawing another red marble are not independent events.

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**Multiplication Rules If A and B are independent events, then**

If A and B are not independent events, where

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Examples Two marbles are drawn from an urn containing 3 red marbles and 2 blue marbles. What’s the probability of drawing two red marbles if the first drawn marble is placed back in the urn?

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Two marbles are drawn from an urn containing 3 red marbles and 2 blue marbles. What’s the probability of drawing two red marbles if the first drawn marble is not placed back in the urn?

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**The Complement of an Event**

The complement, Ec, of event E is the event that E does not occur. Example: If E is the event of rolling an even number on a fair die, then Ec is the event of rolling an even number.

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Complement Rule For any event A, the probability that A does not occur is Sometimes it is easier to use the complement of an event than it is to use the actual event. In this situation you want to use

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Example If you randomly select a number between (and including) 1 and 100, what is the probability that that number is less than 99? If E is the event in this problem, it would be easier to look at Ec, the event of selecting 99 or 100. So in this case,

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This ends the lesson on The Probability of Simple Events

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