The Binomial Distribution -- How to Really Understand it.
Definition:The binomial distribution is a sequence of probabilities with each probability corresponding to the liklihood of a particular event occurring. An event is defined as a sequence of Bernoulli trials that has within it a specific number of successes. The order of success is not important.
e.g.
p_0 corresponds to the probability of an event (sequence) with 0 successes occurring.
p_1 corresponds to the probability of an event (sequence) with 1 success occurring.
etc.
Now, toss a coin 3 times. Each toss is a Bernoulli trial. Consider head to be a success.
***One event is one sequence of 3 coin tosses. Order is inconsequential. i.e., 010 is considered the same sequence as 100***
How many distinct subevents (sequences in which order counts) are possible? ans. 2*2*2 = 8 because there are 2 possibilities for each coin toss (the multiplication rule).
These various possible subevents are 000, 001, 010, 011, 100, 101, 110, 111 where 0 := tail & 1 := head.
Each subevent has a probability of 1/2*1/2*1/2 = 1/8 (the multiplication rule).
Add them up and we get 1/8 + 1/8 +1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 8/8 = 1
BUT WE DON'T CARE ABOUT ORDER; so there is no difference between the three subevents 011, 101, and 110. As far as the binomial distribution is concerned, they are one and the same.
Now,
How many subevents have 0 heads? 1. It is 000.
So the event of 0 heads in 3 tosses has probability ------------------------- p(h=0) = 1/8
How many subevents have 1 head? 3. They are 001, 010, & 100.
So the event of 1 head in 3 tosses has probability -------------------------- p(h=1) = 3/8
How many subevents have 2 heads? 3. They are 011, 101, 110.
So the event of 2 heads in 3 tosses has probability------------------------- p(h=2) = 3/8
How many subevents have 3 heads? 1. It is: 111.
So the event of 3 heads in 3 tosses has probability------------------------- p(h=3) = 1/8 where h = heads.
NOW NOTICE AGAIN ******that we don't care a bit about the order of the successes.******
*** We only care about the total number of successes in each event.***
So the binomial distribution for 3 tosses of a fair coin isp(h=0) = 1/8, p(h=1) = 3/8, p(h=2) = 3/8, p(h=3) = 1/8.
Or we can express the probabilities as follows:
( 3 choose i )[(1/2)^i * 1/2^(3 - i)] , for i = 0, 1, 2, 3
There are two parameters we are concerned with when considering a binomial distribution.
1. p = the probability of a success.
2. n = the number of Bernoulli trials (i.e., the length of the sequence).
One last point. Why do we call this distribution bi -- nomial? It's because each coin toss has precisely two possible outcomes.
1. What is a Bernoulli Trial? PROBLEMS
2. What is a binomial distribution? What are the two parameters with which we are concerned?
3. What is the binomial distribution for four tosses of a fair coin?
4. What is the binomial distribution for three tosses of a coin in which head is considered a success and the probability of getting a head in one toss is 1/4?
5. What is the general expression for the binomial distribution of n Bernoulli Trials with the probability of success = p and the probability of failure = q, where q = 1 - p?
6. What is the trinomial distribution for two tosses of a fair 3-sided coin labelled front, back, and side, in which a toss resulting in the coin landing on its front is considered a success?
7. Do the same for three tosses.
8. Construct a graph for problem no. 4 which plots probabilities vs. the no. of successes (i.e. successes are the horizontal axis (independent variable), and probabilities are the vertical axis (dependent variable).
End of Problems ----- Back to primary page.