Chapter Nine: Binomial Distribution
Definition and Illustration of the Binomial Distribution
– binomial distribution: a probability distribution that results when the following five conditions
are met:
+ 1. there is a series of N trials
+ 2. on each trial, there are only two possible outcomes
+ 3. on each trial, the two possible outcomes are mutually exclusive
+ 4. there is independence between the outcomes of each trial
+ 5. the probability of each possible outcome on any trial stays the same from trial to trial.
When these requirements are met, the binomial distribution tells us each possible outcome of
the N trials and the probability of getting each of these outcomes
– ie: use coin flipping for generating the binomial distribution
+ flip a unbiased penny once
+ two possible outcomes in one flip
+ now we flip two pennies that are unbiased
+ flip of each penny considered a trial → with two pennies, there are two trials (N = 2)
+ possible outcomes of flipping two pennies (Table 9.1)
+ there are four possible outcomes: 2 heads, one head and tail, one tail and head, two tails
– determine the probability of getting each of these outcomes due to chance
+ if chance alone is operating, then each of outcomes is equally likely
+ could've found these probabilities from multiplication and addition rules
+ ie: p(1 head) could've been found from... + now increase N from 2 to 3
+ possible outcomes of flipping three unbiased pennies once are shown in 9.2 – there are eight
possible outcomes
+ distributions resulting from flipping one, two or three fair pennies are shown in Table 9.3 –
they're binomial distributions because they are probability distributions that have been
generated by a situation in which there is a series of trials (N = 1, 2, or 3), where on each trial
there are only two possible outcomes (head or tail) on each trial the possible outcomes are
mutually exclusive, there is independence between trials (there is independence between
outcomes of each coin), and the probability of a head or tail on any trial stays the same from
trial to trial.
+ each distributions gives two pieces of information
- 1. all possible outcomes of N trials
- 2. probability of getting each of the outcomes Generating the Binomial Distribution from the Binomial Expansion
– think about what'd happen when N gets to 15.
+ with 15 pennies, there are (2)^15 = 32,768 different ways that the 15 coins could fall –
binomial expansion
- to generate possible outcomes and associated probabilities we arrived at in previous coin-
flipping experiments, all we need to do is expand the expression (P + Q)^N for the number of
coins in the experiment and evaluate each term in the expansion.
+ ie: if there are two coins, N = 2 and
+ terms P^2, 2P^1Q^1 and Q^2 represent all possible outcomes of flipping two coins at once
– letters of each term (P or PQ or Q) tells us the kinds of events that comprise the outcome, the
exponent of each letter tells us how many of that kind of events there are in the outcome, and
the coefficient of each term tells us how many ways there are of obtaining the outcome
– thus...
+ 1. P^2 indicates that one possible outcome is composed of two P events. The P alone tells us
this outcome is composed entirely of P events. The exponent 2 indicates there are two of this
kind of event. If we associate P with heads, P^2 tells us one possible outcome is two heads
+ 2. 2P^1Q^1 indicates that another possible outcome is one P and one Q event, or one head
and one tail. The coefficient 2 tells us there are two ways to obtain one P and one Q event
+ 3. Q^2 represents an outcome of two Q events or two tails (zero heads).
– The probability of getting each of these possible outcomes is found by evaluating their
respective terms using the terms using the numerical values of P and Q.
+ if coins are fair, then P = Q = 0

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