Binomial, Poisson, and Multinomial Distributions

First, we will talk about binomial probabilities, how to compute their cumulatives, and the mean and standard deviation. Then, we will introduce the Poisson probability formula, define multinomial outcomes, and discuss how to compute probabilities by using the multinomial distribution.


Learning Objectives

  1. Define binomial outcomes
  2. Compute the probability of getting X successes in N trials
  3. Compute cumulative binomial probabilities
  4. 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 probabilities summing to one. These distributions are called binomial distributions.


A Simple Example

The four possible outcomes that could occur if you flipped a coin twice are listed below in Table 1. Note that the four outcomes are equally likely: each has probability 1/4. 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 P(H) and P(H), which is 1/2 x 1/2 = 1/4. The same calculation applies to the probability of a head on Flip 1 and a tail on Flip 2. Each is 1/2 x 1/2 = 1/4.

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 classified 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. Table 2 summarizes the situation.

Number of Heads Probability
0 4-Jan
1 2-Jan
2 4-Jan


Figure 1. Probabilities of 0, 1, and 2 heads.

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.


Source: David M. Lane, https://onlinestatbook.com/2/probability/binomial.html
Public Domain Mark This work is in the Public Domain.