12. The Binomial Probability Distribution
A binomial experiment is one that possesses the following properties:
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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?
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