]> Understanding Basic Statistics: Chapter 6

6 The Binomial Probability Distribution and Related Topics

Now that we are able to calculate probabilities, we can turn our attention to a related, and probably more important topic.

6.1 Introduction to Random Variables and Probability Distributions

A random variable holds the (quantitative) result of a random experiment. It is not like a variable in Algebra—there, in each problem, a variable holds a single, unknown value. A random variable does not hold a single value—each time you repeat the random experiment, the value of the random variable changes (its value is random—hence, random variable).

Since there are two types of quantitative variables, there are also two types of random variables-discrete and continuous. We will look at discrete random variables in this chapter—continuous random variables will come in the next chapter.

Examples:

[1.] Toss three coins, and count the number of coins that land heads up. This is a random experiment (since we get a different number of heads each time), and we are measuring a quantitative variable (number of heads). So we can say that X, the number of heads, is a random variable.

[2.] I toss four dice, and count the number of dice that land on six. This is a random experiment (since we get a different number of sixes each time), and we are measuring a quantitative variable (number of sixes). So we can say that Y, the number of sixes, is a random variable.

[3.] Interview five randomly selected people, and ask each one if they voted in the last presidential election. Count how many say "yes." This is a random experiment (since we ask a randomly selected group of people, and the answers will be different each time), and we are measuring a quantitative variable (number that say "yes"), so we can say that W, the number answering "yes," is a random variable.

Every discrete random variable has a probability distribution—a list of the possible values that can be measured, and how often each of those values occurs. The probability distribution for a discrete random variable is often given as a table. Since we now know how to calculate probability, we can (usually) determine how often each value of the variable occurs. Fortunately, we will probably not need to calculate the probabilities (yet)—they will usually be given to us (for now).

Examples:

[4.] Consider the random variable X from the earlier examples. What kinds of values can X take? Since it is measuring the number of heads (out of 3 coins), X can take on the values 0, 1, 2, or 3. Here's the probability distribution for X:

# of Heads (x)	Probability (P(x))
0	0.125
1	0.375
2	0.375
3	0.125

[5.] Consider the random variable Y from the earlier examples. What kinds of values can Y take? Since this is measuring the number of sixes, you can have anywhere from none to all of the dice coming up six—so the possible values are 0 through 4. Here's the probability distribution:

# of Sixes (y)	Probability (P(y))
0	0.4823
1	0.3858
2	0.1157
3	0.0154
4	0.0008

[6.] For a fundraiser, a club is holding a raffle. 1000 raffle tickets are sold at $1 each. At the end of the fundraiser, the club will randomly select one ticket as the winner—the person holding the winning ticket wins $100 in cash. Suppose that you have one ticket. Let R be a random variable that holds your profit (money won - money spent). What does the probability distribution of R look like?

There are only two possible values of R—either you win ($100 - $1 = $99), or you lose (0$ - $1 = -$1). The probability that you win is one in a thousand, or 0.001; so the probability that you lose is 0.999. Here's the distribution:

Profit (r)	Probability (P(r))
$99	0.001
-$1	0.999

Since a probability distribution represents all possible outcomes, the probabilities should add to 1 (or 100%).

There are two things that we need to be able to do with a probability distribution—find the mean and the standard deviation. Here are the formulas:

Equation 1 - Mean of a Discrete Random Variable

$μ_{X} = \sum (x \cdot P (x))$

Equation 2 - Standard Deviation of a Random Variable

$σ_{X} = \sqrt{\sum [{(x - μ)}^{2} \cdot P (x)]}$

We used $\overline{x}$ for the mean in Chapter 3, but that was for a sample—just some of the possible values. Now that we're talking about all possible values, we change the symbol to the lowercase greek letter mu. Similarly, the standard deviation changes from s to the greek lower case sigma.

Now, we're not actually going to use these formulas—we'll let the calculator do it for us!

Put the first column (values of the variable) into L₁. Put the second column (probabilities) into L₂. Now, enter: 1-Var Stats L₁, L₂

The calculator doesn't know that this is a probability distribution, so it will put the mean (μ) into $\overline{x}$ . The standard deviation will, however, be in its proper place (σ).

If you want to find μ and σ without the calculator, then feel free to read the examples in the textbook!

Examples:

[7.] For our random variable X, the mean is 1.5 and the standard deviation is 0.8660.

[8.] For our random variable Y, the mean is 0.667 and the standard deviation is 0.7454.

[9.] For our random variable R, the mean is -$0.9 and the standard deviation is $3.1607.

The mean of a discrete random variable is also called the expected value—this is because it represents the average result (over many, many trials). So, for example, in the case of the raffle, a player can expect to lose $0.90 (90¢) on average. If a person were to buy one ticket in lots of these raffles, then that person will lose an average of 90¢ per raffle.

6.2 Binomial Probabilities

There are many different types of discrete random variables, but there is one that is more important than most—the binomial.

The Binomial Experiment

A binomial experiment is one where [a] each trial results in either success or failure; [b] where the probability of success (p) remains the same for each trial; and [c] where each trial is independent of the others.

If your random variable counts the number of successes in a fixed number of trials (n) of a binomial experiment, then the variable is called a binomial random variable.

Examples:

[10.] Toss three coins, and count the number of coins that land heads up. Each coin is a binomial experiment, since it either results in success (heads) or failure (tails); each trial has the same probability of success (50%); and the results of each toss are independent of every other toss. Since our random variable X counts the number of successes (number of heads) in a fixed number of trials (3), the variable X is a binomial random variable.

[11.] I toss four dice, and count the number of dice that land on six. Each die is a binomial experiment, since it either results in success (6) or failure (1 through 5); each trial has the same probability of success ( $\frac{1}{6}$ ); and since each die roll is independent of the others. Since our random variable Y counts the number of successes (sixes) in a fixed number of trials (4), the variable Y is a binomial random variable.

The Binomial Formula

So—how do you find the probability that a binomial random variable will take on a certain value? Why, the binomial formula, of course! In this formula, X is the binomial random variable, r is the number of successes that you are wondering about, n is the number of trials, p is the probability of success on any given trial, and q is the probability of failure (1 - p) on any given trial.

Equation 3 - The Binomial Formula

$P (X = r) = (\binom{n}{r}) p^{r} q^{n - r}$

Once again, we won't be doing this by hand—we'll let the calculator do it for us! The command is buried deeply. 2nd VARS is the Distribution menu; near the bottom of the list is the command binompdf(. This is the binomial formula—just enter n, p and r (in that order) after the left parenthesis.

Examples:

[12.] Toss three coins, and count the number of coins that land heads up. What is the probability that exactly two of them are heads? P(X = 2) = binompdf(3,0.5,2) = 0.375.

[13.] I toss four dice, and count the number of dice that land on six. What is the probability that exactly three of them land on six? P(Y = 3) = binompdf(4,1/6,3) = 0.0154.

[14.] I toss four dice, and count the number of dice that land on six. What is the probability that more than 2 land on six? Notice that this is a little different—instead of asking about a single number, it's asking about a set of numbers (more than 2 means exactly 3 or exactly 4). So the question is asking P(Y > 2), which is the same as P(Y = 3 OR Y = 4). We saw how to deal with these OR problems in Chapter 5; thus, P(Y = 3 OR Y = 4) = P(Y = 3) + P(Y = 4) = binompdf(4,1/6,3)+binompdf(4,1/6,4) = 0.0162.

Note the table that translates English into Math on page 233! This could have been helpful on the last example.

6.3 Additional Properties of the Binomial Distribution

The Shape of the Graph

It is possible to construct a histogram of a binomial distribution. Perhaps we'll see how to do that in class…

The Mean of the Binomial

You don't have to use a big, ugly formula to find the mean of a binomial random variable—there is a shorter way! If B is a binomial random variable, then the following formula gives the mean.

Equation 4 - Mean of a Binomial Random Variable

$μ_{B} = np$

Examples:

[15.] Toss three coins, and count the number of coins that land heads up (X). The mean is $μ_{X} = (3) (0.5) = 1.5$ . So, if we toss three coins lots of times, then we can expect to get an average of 1.5 heads per try.

[16.] I toss four dice, and count the number of dice that land on six (Y). The mean is $μ_{Y} = (4) (\frac{1}{6}) = \frac{2}{3}$ . Thus, if we roll four dice many different times, we can expect to get an average of $\frac{2}{3}$ (less than one six in every try) per try.

The Standard Deviation of the Binomial

There is also a quick formula for the standard deviation of a binomial random variable:

Equation 5 - Standard Deviation of a Binomial Random Variable

$σ_{B} = \sqrt{n \cdot p \cdot q}$

(remember that q = 1 - p).

Examples:

[17.] Toss three coins, and count the number of coins that land heads up (X). The standard deviation is $σ_{X} = \sqrt{(3) (0.5) (0.5)} = \sqrt{\frac{3}{4}}$ .

[18.] I toss four dice, and count the number of dice that land on six (Y). The standard deviation is $σ_{Y} = \sqrt{(4) (\frac{1}{6}) (\frac{5}{6})} = \sqrt{\frac{5}{9}}$ .

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