Binomial Probability

P (x) = C (n, x) p^{x} (1 - p)^{n - x}

Probability of exactly x successes in n independent Bernoulli trials with success probability p. C(n,x) = n! / (x!(n−x)!). Mean μ = np and variance np(1−p). Bernoulli is the n = 1 case.

$n$: Number of trials
$x$: Number of successes
$p$: Success probability per trial
$C (n, x)$: Binomial coefficient n choose x

Step-by-step solved example

n = 5, p = 0.4. Find P(X = 2).

Binomial counts become normal-like for large n.

1. Coefficient
C(5,2) = 10.
$C (5, 2) = 10$
2. Probability
P = 10 (0.4)² (0.6)³ = 10×0.16×0.216 = 0.3456.
$P = 0.3456$

Answer: P = 0.346

10 practice questions

0/10 correct

1.C(5,2) equals

2.C(n,0) equals

3.Fair coin, 2 tosses, P(2 heads) is

4.Mean of a binomial random variable is

5.n = 10, p = 0.3. Mean equals

6.Variance np(1−p) for n = 4, p = 0.5 is

7.C(6,6) equals

8.n = 3, p = 0.5, P(X = 1) equals

9.A Bernoulli trial is binomial with

10.Σ P(x) from x = 0 to n equals