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)=C_x^n p^x q^(n-x)`
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)
`C_x^n` 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.
A die is tossed `3` times. What is the probability of
(a) No fives turning up?
(b) `1` five?
(c) `3` fives?
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?
In the old days, there was a probability of `0.8` of success in any attempt to make a telephone call. (This often depended on the importance of the person making the call, or the operator's curiosity!)
Calculate the probability of having `7` successes in `10` attempts.
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?
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`.]
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?