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Binomial Distribution

Module by: David Lane. E-mail the author

When you flip a coin, there are two possible outcomes: heads and tails. Each outcome has a fixed probability, the same from trial to trial. In the case of coins, heads and tails each have the same probability of 1/2. More generally, there are situations in which the coin is biased, so that heads and tails have different probabilities. In the present section, we consider probability distributions for which there are just two possible outcomes with fixed probability summing to one. These distributions are called are called binomial distributions.

A Simple Example

The four possible outcomes that could occur if you flipped a coin twice are listed in Table 1. Note that the four outcomes are equally likely: each has probability 1/4 14. To see this, note that the tosses of the coin are independent (neither affects the other). Hence, the probability of a head on Flip 1 and a head on Flip 2 is the product of PrH H and PrH H , which is 1/2×1/2=1/4 12 12 14 . The same calculation applies to the probability of a head on Flip one and a tail on Flip 2. Each is 1/2×1/2=1/4 12 12 14 .

Table 1: Four Possible Outcomes
Outcome First Flip Second Flip
1 Heads Heads
2 Heads Tails
3 Tails Heads
4 Tails Tails

The four possible outcomes can be classifid in terms of the number of heads that come up. The number could be two (Outcome 1), one (Outcomes 2 and 3) or 0 (Outcome 4). The probabilities of these possibilities are shown in Table 2 and in Figure 1. Since two of the outcomes represent the case in which just one head appears in the two tosses, the probability of this event is equal to 1/4+1/4=1/2 14 14 12 . Table 1 summarizes the situation.

Table 2: Probabilities of Getting 0,1, or 2 heads.
Number of Heads Probability
0 1/4
1 1/2
2 1/4
Figure 1: Probabilities of 0, 1, and 2 heads.
Figure 1 (binomial1.gif)

Figure 1 is a discrete probability distribution: It shows the probability for each of the values on the X-axis. Defining a head as a "success," Figure 1 shows the probability of 0, 1, and 2 successes for two trials (flips) for an event that has a probability of 0.5 of being a success on each trial. This makes Figure 1 an example of a binomial distribution.

The Formula for Binomial Probabilities

The binomial distribution consists of the probabilities of each of the possible numbers of successes on NN trials for independent events that each have a probability of π(the Greek letter pi) of occurring. For the coin flip example, N=2 N 2 and π=0.5 0.5 . The formula for the binomial distribution is shown below: Prx=N!x!⁢(N−x)!⁢πx⁢1−πN−x x N x N x x 1 N x where Prx x is the probability of x x successes out of NN trials, NN is the number of trials, and πis the probability of success on a given trial. Applying this to the coin flip example, Pr0=2!0!⁢(2−0)!⁢0.50⁢1−0.52−0=22⁢1×.25=0.25 0 2 0 2 0 0.5 0 1 0.5 2 0 2 2 1 .25 0.25 Pr1=2!1!⁢(2−1)!⁢0.51⁢1−0.52−1=21⁢.5×.5=0.50 1 2 1 2 1 0.5 1 1 0.5 2 1 2 1 .5 .5 0.50 Pr2=2!2!⁢(2−2)!⁢0.502⁢1−0.52−2=22⁢.25×1=0.25 2 2 2 2 2 0.5 02 1 0.5 2 2 2 2 .25 1 0.25 If you flip a coin twice, what is the probability of getting one or more heads? Since the probability of getting exactly one head is 0.50 and the probability of getting exactly two heads is 0.25, the probability of getting one or more heads is 0.50+0.25=0.75 0.50 0.25 0.75 .

Now suppose that the coin is biased. The probability of heads is only 0.4. What is the probability of getting heads at least once in two tosses? Substituting into our general formula above, you should obtain the answer .64.

Cumulative Probabilities

We toss a coin 12 times. What is the probability that we get from 0 to 3 heads? The answer is found by computing the probability of exactly 0 heads, exactly 1 head, exactly 2 heads, and exactly 3 heads. The probability of getting from 0 to 3 heads is then the sum of these probabilities. The probabilities are: 0.0002, 0.0029, 0.0161, and 0.0537. The sum of the probabilities is 0.073. The calculation of cumulative binomial probabilities can be quite tedious. Therefore we have provided a binomial calculator to make it easy to calculate these probabilities.

Binomial Calculator:

Click here for the binomial calculator.

Mean and Standard Deviation of Binomial Distributions

Consider a coin-tossing experiment in which you tossed a coin 12 times and recorded the number of heads. If you performed this experiment over and over again, what would the mean number of heads be? On average, you would expect half the coin tosses to come up heads. Therefore the mean number of heads would be 6. In general, the mean of a binomial distribution with parameters NN (the number of trials) and π (the probability of success for each trial) is: m=N⁢π m N where m m is the mean of the binomial distribution. The variance of the binomial distribution is: s2=N⁢π⁢(1−π) s 2 N 1 where s2 s 2 is the variance of the binomial distribution.

Let's return to the coin tossing experiment. The coin was tossed 12 times so N=12 N 12 . A coin has a probability of 0.5 of coming up heads. Therefore, π=0.5 0.5 . The mean and standard deviation can therefore be computed as follows: m=N⁢π=12×0.5=6 m N 12 0.5 6 s2=N⁢π⁢(1−π)=12×0.5×(1.0−0.5)=3.0 s 2 N 1 12 0.5 1.0 0.5 3.0 Naturally, the standard deviation s s is the square root of the variance s2 s 2 .

Binomial Calculator

Glossary

binomial distributions:
A probability distribution for independent events for which there are only two possible outcomes such as a coin flip. If one of the two outcomes is defined as a success, then the probability of exactly x x successes out of N N trials (events) is given by: Prx=N!x!⁢(N−x)!⁢πx⁢1−πN−x x N x N x x 1 N x where π is the probability of success one one trial.
conditional probability:
The probability that event A occurs given that event B has already occurred is called the conditional probability of A given B. Symbolically, this is written as PrA| B B A . The probability it rains on Monday given that it rained on Sunday would be written as Pr⁢ Rain on Sunday Rain on Monday Pr Rain on Sunday Rain on Monday .
continuous variables:
Variables that can take on any value in a certain range. Time and distance are continuous; gender, SAT score and "time rounded to the nearest second" are not. Variables that are not continuous are known asdiscrete variables. No measured variable is truly continuous; however, discrete variables measured with enough precision can often be considered continuous for practical purposes.
discrete:
Variables that can only take on a finite number of values are called "discrete variables." All qualitative variables are discrete. Some quantitative variables are discrete, such as performance rated as 1,2,3,4, or 5, or temperature rounded to the nearest degree. Sometimes, a variable that takes on enough discrete values can be considered to be continuous for practical purposes. One example is time to the nearest millisecond. Variables that can take on an infinite number of possible values are called continuous variables.
independent events:
Intuitively, two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other. For example, if you throw two dice, the probability that the second one comes up 1 is independent of whether the first die came up 1. Formally, this can be stated in terms of conditional probabilities: PrA| B =PrA B A A and PrB| A =PrB A B B .
levels of measurement:
Measurement scales differ in their level of measurement. There are four common levels of measurement:
  1. Nominal scales are only labels.
  2. Ordinal Scales are ordered but are not truly quantitative. Equal intervals on the ordinal scale do not imply equal intervals on the underlying trait.
  3. Interval scales are are ordered and equal intervals equal intervals on the underlying trait. However, interval scales do not have a true zero point.
  4. Ratio scales are interval scales that do have a true zero point. With ratio scales, it is sensible to talk about one value being twice as large as another, for example.
nominal scale:
A nominal scale is one of four Levels of Measurement. No ordering is implied, and addition/subtraction and multiplication/division would be inappropriate for a variable on a nominal scale. FemaleMale Female Male and BuddhistChristianHinduMuslim Buddhist Christian Hindu Muslim have no natural ordering (except alphabetic). Occasionally, numeric values are nominal: for instance, if a variable was coded as Female=1, Male=2, the set 12 1 2 is still nominal.
ordinal scale:
One of four levels of measurement, an ordinal scale is a set of ordered values. However, there is no set distance between scale values. For instance, for the scale: (Very Poor, Poor, Average, Good, Very Good) is an ordinal scale. You can assign numerical values to an ordinal scale: rating performance such as 1 for "Very Poor," 2 for "Poor," etc, but there is no assurance that the difference between a score of 1 and 2 means the same thing as the difference between a score of and 3.
probability distribution:
For a discrete random variable, a probability distribution contains the probability of each possible outcome. The sum of all probabilities is always 1.0.
qualitative variables:
Categorical Variable: Also known as categorical variables, qualitative variables are variables with no natural sense of ordering. For instance, hair color (Black, Brown, Gray, Red, Yellow) is a qualitative variable, as is name (Adam, Becky, Christina, Dave . . .). Qualitative variables can be coded to appear numeric but their numbers are meaningless, as in male=1, female=2. Variables that are not qualitative are known as quantitative variables.
quantitative variables:
Variables that have are measured on a numeric or quantitative scale. Ordinal, interval and ratio scales are quantitative. A country's population, a person's shoe size, or a car's speed are all quantitative variables. Variables that are not quantitative are known as qualitative variables.
ratio scale:
One of the four basic levels of measurement, a ratio scale is a numerical scale with a true zero point and in which a given size interval has the same interpretation for the entire scale. Weight is a ratio scale, Therefore it is meaningful to say that a 200 pound person weighs twice as much as a 100 pound person.
variables:
Something that can take on different values. For example, different subjects in an experiment weight different amounts. Therefore "weight" is a variable in the experiment. Or, subjects may be given different doses of a drug. This would make "dosage" a variable. Variables can be dependent or independent, qualitative or quantitative, and continuous or discrete.

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