9-7 Functions of a Random Variable

This section is aimed at finding all of the probabilities of the possible events in a random experiment. EXCITING!

The random experiments in this section consist of repeating the experiment several times. The experiments also have only two possible outcomes.





In this equation,
N is the total times that an experiment is being done.
X is the number of times the event you want occurs in N repetitions.
A is the probability that the event we want does not occur in the N repetitions.
B is the probability that the event we want does occur in the N repetitions.

Because solving these types of equations shows the probabilities of all the possible outcomes, it is called a probability distribution.

The expressions that we are finding are terms in a binomial series. We get this series from expanding certain terms. For example, if the two probabilities we are using are .6 and .4, and we are repeating the experiment 5 times, we would be expanding the equation: (.6+.4)^5.

Since the probabilities apply to binomial series, we are specifically finding the binomial distribution. And, because there are only two possible outcomes, each trial is called a binomial experiment.

Example:
A butane candle lighter does not always light when you pull the trigger. Suppose that a lighter has a 60% probability of lighting on any one pull. You pull the triggier six times. Let P(x) be the probability it lights exactly x of those times. Show how to calculate P(4).

To solve this equation, all we have to do is plug the numbers into the equation!

N would be 6 because that is the total number of times we are pulling the trigger.
X would be 4 because we want to find out the probability of the lighter lighting 4 of the 6 times.
A would be .4 because that is the probability the lighter will not light.
B would be .6 because that is the probability the lighter will light.

So...
P(x)=6C4 * (.4^2)*(.6^4)
When we plug the equation into our calculator, we get
P(x)=.31104

There is a 31.1% chance that the lighter will light four of the six times.

Now, there is an easy way to calculate the probabilities for the rest of the outcomes. On your calculator, go STAT then EDIT. When the top of the L2 column is highlighted, press 2nd VARS. Go to binompdf. Inside the parenthesis, type in 6 (the total number of times we are performing the experiment) followed by a comma and then .6 (the probability the lighter will light).
You should get the values for all of the outcomes of the experiment.

Thanks to our handy dandy calculators, we can easily find all of the values in the binomial distribution, hooray!

This website is a little confusing because they use complicated notation, but if you study it long enough it makes sense.

http://mathworld.wolfram.com/BinomialDistribution.html

I have been tap dancing for most of my life, so here is a link to one of my favorite tap dancing videos...it's the awesome stair dance with Shirley Temple and Bill Bojangles!

http://youtube.com/watch?v=ImRGu4kjuZY

Gina, you're up next!