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Probability

Page history last edited by Andrea Grieser 13 years, 8 months ago

Probability is the measure of the likelihood an event is to occur.  We find the probability of an event by taking the ratio of number of favorable outcomes to the number of possible outcomes.

For example, if we want to know the probability of rolling a 4 on a six-sided number cube, we would determine the number of ways to roll as 4, our favorable outcome, which is 1 (there is only one 4 on a six-sided number cube).  This is our numerator.  Then we determine the number of possible outcomes, which is 6, since there are 6 sides to the number cube.  So our probability of rolling a 4 would be 1/6.

 

We would write this as:  P(4) = 1/6

 

Another example:  what is the probability of choosing a red candy in a bag that contains 4 red candies and 6 blue candies?

 

The number of favorable outcomes is 4, since there are 4 red candies.  The number of possible outcomes is 10, since there are 10 total candies in the bag.  Therefore the probability of drawing a red candy is 4/10, which in simples form is 2/5.  P(red) = 2/5

 

In counting outcomes, it is sometimes helpful to create tree diagrams or apply the fundamental counting principle

 

Experimental vs. Theoretical Probability

The process above describes theoretical probability.  This is what mathematically would expect to happen.  Experimental probability describes what actually happens in actually performing the event.  If we to actually draw candies from the bag in the example above (putting back the candy after we draw it), each draw is called a trial.  If we perform 5 trials, we may not see a probability of 2/5.  However, if we perform 100 trials, we would most likely see the probability getting closer to 2/5.  If we performed an infinite number of trials, we expect the probability to be 2/5, our theoretical probability.

 

Compound Events

Compound events consist of two or more events. 

 

Mutually exclusive events do not occur at the same time - either one or the other will occur.  For example, if I am rolling a six-sided number cube, rolling an odd number and a 2 are mutually exclusive - either one or the other will occur.  To find the probability of mutually exclusive compound events, we add the probability of each individual event.

 

In this example we want to know P(odd or 2).  P(odd) = 3/6 = 1/2, since there are 3 odd number (1, 3, 5) on a six-sided number cube.  P(2) = 1/6, since there is only one way to roll a 2. 

So P(odd or 2) = P(odd) + P(2) = 3/6 + 1/6 = 4/6 = 2/3

 

The formula for finding the probability of mutually exclusive events is: 

P(A or B) = P(A) + P(B)

  

Independent and dependent compound events occur when we have a first event AND then one more additional events.  When the outcome of one event does not influence the outcome of another, we have an independent compound event.  To find the probability of independent compound events, we multiply the probabilities of each individual event.

 

In the above example, where we have a bag of 10 candies with 4 of them red and 6 of them blue, suppose we want to find the probability of first drawing a red candy and then drawing a blue candy, if we put back whatever we drew the first time before drawing the second time.

 

We want to know: P(red) and P(blue), with replacement (this means we will be putting the candy back after the first draw)

 

P(red) = 2/5 (as we calculated above) and P(blue) = 3/5, since there are 6 out of 10 blue candies.

To find P(red) and P(blue) we multiply:  2/5*3/5 = 6/25

 

The formula for finding the probability of independent events is: 

P(A and B, with replacement) = P(A) * P(B)

 

If we do NOT replace the candy, we have a dependent compound event.  We still multiply the probability of the events, but the probability of the second event depends on the outcome of the first event, because we will not be putting back the candy drawn, so the contents of the bag of candy will be different.

 

We want to know:  P(red) and P(blue), without replacement (we will not be putting the candy back)

 

P(red) is still 2/5, since there are 4 red out of 10 candies when we pick the first candy.  However, after we pick out a red candy (and don't put it back - maybe we ate it!), the bag now contains 3 red candies and 6 blue candies, for a total of 9 candies.  So now P(blue) will be 6/9 = 2/3.  Therefore P(red) and P(blue) without replacement = 2/5 * 2/3 = 4/15

 

The formula for finding the probability of dependent events is: 

P(A and B, without replacement) = P(A)*P(B following A)

 

These class notes provide further explanation and examples on probability in general, while these class notes provide more information on compound events and finding odds

 

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