Concept of Binomial Probabilities

Binomial Probabilities:

Binomial Experiment:

Any experiment which satisfies the given four conditions is termed as binomial experiment.

•    Fixed number of trials
•    Each and every trial is independent of others
•    There are just two outcomes
•    The probability of each outcome stays constant from trial to trial.

These can be concluded as: Any experiment with a fixed number of independent trials, all of which can only encompass two possible outcomes.

The fact which each trial is independent really means that the probabilities stay constant.

Illustrations of binomial experiments:

•    Flipping a coin 20 times to see how many tails take place.
•    Asking 300 people if they watch ABC news.
•    Rolling a die to see if 5 appears.

Illustrations that are not binomial experiments:

•    Rolling a die until a 6 appears (that is, not a fixed number of trials)
•    Asking 30 people how old they are (that is, not two outcomes)
•    Drawing 5 cards from a deck for a poker hand (done with no replacement, thus not independent)

Binomial Probability Function:

Find out the probability of rolling exactly two sixes in 6 rolls of a die?

There are mainly five things you require to do work a binomial story problem.

a) At first define Success. Success should be for a single trial. Success = ‘Rolling a 6 on a single die’.

b) Define the probability of success (p): p = 1/6

c) Find out the probability of failure: q = 5/6

d) Define the number of trials: n = 6

e) Define the number of successes out of such trials: x = 2

At any time a six appears, it is a success (represented by S) and anytime something else appears, this is a failure (represented by F). The ways you can exactly get 2 successes in 6 trials are shown below. The probability of each is written to right of the way it could take place. Since the trials are independent, the probability of event (all six dice) is the product of each probability of each outcome (of die).

 1 FFFFSS  5/6 * 5/6 * 5/6 * 5/6 * 1/6 * 1/6 = (1/6)^2 * (5/6)^4
 2 FFFSFS  5/6 * 5/6 * 5/6 * 1/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
 3 FFFSSF  5/6 * 5/6 * 5/6 * 1/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
 4 FFSFFS  5/6 * 5/6 * 1/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
 5 FFSFSF  5/6 * 5/6 * 1/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
 6 FFSSFF  5/6 * 5/6 * 1/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
 7 FSFFFS  5/6 * 1/6 * 5/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
 8 FSFFSF  5/6 * 1/6 * 5/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
 9 FSFSFF  5/6 * 1/6 * 5/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
10 FSSFFF  5/6 * 1/6 * 1/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
11 SFFFFS  1/6 * 5/6 * 5/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
12 SFFFSF  1/6 * 5/6 * 5/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
13 SFFSFF  1/6 * 5/6 * 5/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
14 SFSFFF  1/6 * 5/6 * 1/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
15 SSFFFF  1/6 * 1/6 * 5/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4

Note that each of 15 probabilities are exactly similar: (1/6)^2 * (5/6)^4.

Also, note that 1/6 is the probability of success and you required 2 successes. The 5/6 is a probability of failure, and if 2 of 6 trials were success, then 4 of 6 should be failures. It will be noted that 2 is the value of x and 4 is the value of n-x.

Moreover note that there are fifteen ways this can take place. This is the number of ways 2 successes can be occur in 6 trials with no repetition and order not being significant, or a combination of 6 things, 2 at a time.

The probability of obtaining exactly x success in n trials, with the probability of success on single trial being p is:

P(X=x) = nCx * p^x * q^(n-x)

Illustration: A coin is flipped 10 times. Find out the probability that precisely 6 heads will occur.

a) Success = ‘A head is tossed on a single coin’.
b) p = 0.5
c) q = 0.5
d) n = 10
e) x = 6

P(x=6) = 10C6 * 0.5^6 * 0.5^4 = 210 * 0.015625 * 0.0625 = 0.205078125

Mean, Variance and Standard Deviation:

The mean, variance and standard deviation of the binomial distribution are extremely simple to find.

μ = np
σ2 = npq
σ = √npq

Illustration: Determine the mean, variance and standard deviation for the number of sixes which appear when rolling 30 dice.
Success = ‘a six is rolled on a single die’. p = 1/6, q = 5/6.

The mean is equal to 30 * (1/6) = 5. The variance is 30 * (1/6) * (5/6) = 25/6. The standard deviation is the square root of variance = 2.041241452.

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