David M. Lane
Basic Probability, Variability
- Define binomial outcomes
- Compute probability of getting X successes in N trials
- Compute cumulative binomial probabilities
- Find the mean and standard deviation of a binomial distribution
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
The Formula for Binomial Probabilities
The binomial distribution consists of the probabilities
of each of the possible numbers of successes on N trials for
independent events that each have a probability of π (the
Greek letter pi) of occurring. For the coin flip example, N
= 2 and π = 0.5.
The formula for the binomial distribution is shown below:
where P(x) is the probability of x successes out
of N trials, N is the number of trials, and π is the probability
of success on a given trial.
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.
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.
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.
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 N (the number of trials) and π (the probability
of success for each trial) is:
μ = Nπ
where μ is the mean of the binomial distribution.
The variance of the binomial distribution is:
σ2 = Nπ(1-π)
where σ2 is the
variance of the binomial distribution.
Let's return to the coin tossing experiment. The
coin was tossed 12 times so N = 12. A coin has a probability of
0.5 of coming up heads. Therefore, π = 0.5. The mean and standard
deviation can therefore be computed as follows:
μ = Nπ= (12)(0.5) = 6
σ2 = Nπ(1-π)= (12)(0.5)(1.0
- 0.5) = 3.0.
Naturally, the standard deviation (σ) is
the square root of the variance (σ2).
Please answer the questions: