4.2. Binomial Distributions http://www.ck12.org
- What is the probability of flipping exactly two heads when a coin is flipped ten times?
- What is the probability of rolling a 2 exactly twice in 15 rolls of a fair die?
There are a few important characteristics of a binomial experiment. First, there must be only two possible outcomes
of each trial. The probability of success in each trial must be the same. The results of each trial must be independent
of one another.
Flipping a coin has only two outcomes (heads and not heads). Each outcome has a 50% chance of success. Each
coin flip is independent of previous coin flips. If I observe 4 heads in a row, the probability of the next flip resulting
in a heads is still 50%.
Verifying these three conditions is important for helping us identify binomial experiments. Once we have a binomial
experiment and we can identify a few pieces of information (like n, a, p and q), then we can use the general formula
for finding the probability of each possible outcome.
There are several reasons why we want to calculate each possible outcome. We can chart the probabilities of
the different outcomes in a distribution. This will allow us to to identify patterns and compare the different
probabilities visually. This helps with making predictions about the outcomes of future experiments and gives
additional information about how many trials would be necessary to draw useful conclusions.
In previous Concepts, you did a little work generating the formula used to calculate probabilities for binomial
experiments. Here is the general formula for finding the probability of a binomial experiment.
The probability of gettingXsuccesses in n trials is given by:
P(X=a) =nCa×pa×q(n−a)
where:
- a is the number of successes from the trials.
- p is the probability of success.
- q is the probability of failure.
One of the reasons why we study binomial distributions is because they use discrete data to approximate a normal
distribution which focuses on continuous data. The more trials there are in the experiment, the better this approxi-
mation is.
Example A
What is the probability of rolling a 2 exactly four times when rolling a fair die 10 times?
n= 10
a= 4p=1
6
q=5
6
P(X=a) =nCa×pa×q(n−a)P(4 twos) = 10 C 4 ×