12. The Binomial Probability Distribution
A binomial experiment is one that possesses the following properties:
On this page...
The experiment consists of n repeated trials;
Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial);
The probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent.
The number of successes X in n trials of a binomial experiment is called a binomial random variable.
The probability distribution of the random variable X is called a binomial distribution, and is given by the formula:
P(X) = Cnxpxqn−x
where
n = the number of trials
x = 0, 1, 2, ... n
p = the probability of success in a single trial
q = the probability of failure in a single trial
(i.e. q = 1 − p)
Cnx is a combination
P(X) gives the probability of successes in n binomial trials.
Mean and Variance of Binomial Distribution
If p is the probability of success and q is the probability of failure in a binomial trial, then the expected number of successes in n trials (i.e. the mean value of the binomial distribution) is
E(X) = μ = np
The variance of the binomial distribution is
V(X) = σ2 = npq
Note: In a binomial distribution, only 2 parameters, namely n and p, are needed to determine the probability.
EXAMPLE 1
A die is tossed 3 times. What is the probability of
(a) No fives turning up?
(b) 1 five?
(c) 3 fives?
EXAMPLE 2
Hospital records show that of patients suffering from a certain disease, 75% die of it. What is the probability that of 6 randomly selected patients, 4 will recover?
EXAMPLE 3
In the old days, there was a probability of 0.8 of success in any attempt to make a telephone call.
Calculate the probability of having 7 successes in 10 attempts.
EXAMPLE 4
A (blindfolded) marksman finds that on the average he hits the target 4 times out of 5. If he fires 4 shots, what is the probability of
(a) more than 2 hits?
(b) at least 3 misses?
EXAMPLE 5
The ratio of boys to girls at birth in Singapore is quite high at 1.09:1.
What proportion of Singapore families with exactly 6 children will have at least 3 boys? (Ignore the probability of multiple births.)
[Interesting and disturbing trivia: In most countries the ratio of boys to girls is about 1.04:1, but in China it is 1.15:1.]
EXAMPLE 6
A manufacturer of metal pistons finds that on the average, 12% of his pistons are rejected because they are either oversize or undersize. What is the probability that a batch of 10 pistons will contain
(a) no more than 2 rejects? (b) at least 2 rejects?
Didn't find what you are looking for on this page? Try search:
The IntMath Newsletter
Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!
Probability Lessons on DVD
Easy to understand probability lessons on DVD. See samples before you commit.
More info: Probability videos
Book mark this page
Add this page to Del.icio.us, Furl, Digg, StumbleUpon, Google, whatever...
Need a break? Play a math game. Well, they all involve math... No, really!
SNB files
