Binomial Experiment

Binomial Experiment

How to find the probability of the binomial experiments: formula, examples, and their solutions.

Formula

P(X = k) = n_C_k * p^k * q^(n - k). n: Number of trials, k: Number of 'want' happened, p: Probability of 'want', q = 1 - p.

Think about an independent event
whose probability of 'want' is p.
(The probability of 'not want' is 1 - p = q.)

Probability of (not A, Complement)

For n trials,
if the 'want' happens k times,
('not want' happens 'n - k' times)
then that probability is nCkpkqn - k.

It's like finding the ways of
choosing p k times
and choosing q 'n - k' times.

Binomial theorem

Each probability is multiplied (p or q)
because each trial is an independent event.

Example 1

A coin is tossed 6 times. Find P(3 'head's).

Tossing a coin is an independent event.
And it happens 6 times.

So this is a binomial experiment.

n = 6 (6 trials)
k = 3 (3 'head's)
p = 1/2 (Probability of 'head' shown)
q = 1 - 1/2 = 1/2 (Probability of 'head' not shown)

Use the binomial experiment formula.

P = 6C3⋅(1/2)3⋅(1/2)6 - 3

Cancel 6 and 2⋅3.

Cancel 4 and change 23 to 2.

Example 2

For the given spinner, when spinning it 5 times, find P(2 'blue's). The measure of the blue part's interior angle: 120.

Spinning a spinner is an independent event.
And it happens 5 times.

So this is a binomial experiment.

n = 5 (5 trials)
k = 2 (2 'blue's)
p = 1/3 (Probability of the 'blue' part)
q = 1 - 1/3 = 2/3 (Probability of the other part)

Use the binomial experiment formula.

P = 5C2⋅(1/3)2⋅(2/3)5 - 2

Example 3

A fair die is tossed 3 times. Find P(at most one '6').

Tossing a fair die is an independent event.
And it happens 3 times.

So this is a binomial experiment.

n = 3 (3 trials)
k = 0, 1 (0 or 1 '6')
p = 1/6 (Probability of '6' shown)
q = 1 - 1/6 = 5/6 (Probability of '6' not shown)

Use the binomial experiment formula.

P = 3C0⋅(1/6)0⋅(5/6)3 - 0 [P(zero '6')]
+ 3C1⋅(1/6)1⋅(5/6)3 - 1 [P(one '6')]

Find the prime factorizations of 200 and 216.
Then cancel the common factors.