Maharashtra State Board · Class 12 · Mathematics · Part II Chapter 8

Binomial Distribution — Formula Sheet

Board Formulas

14 formulas

1.Bernoulli Trial
$P (success) = p, P (failure) = q, p + q = 1$
$p$ : Probability of success in one trial · $q$ : Probability of failure in one trial, 1 − p
A trial with exactly two outcomes. For a sequence of Bernoulli trials, the trials are independent and p stays the same from trial to trial.
2.Bernoulli Distribution (one trial)
$P (X = 1) = p, P (X = 0) = q; E (X) = p, Var (X) = pq$
X counts successes in a single trial. It is the binomial distribution with n = 1.
3.Binomial Distribution Notation
$X \sim B (n, p), X = 0, 1, 2, \dots, n$
X is the number of successes in n Bernoulli trials. The two parameters n and p fix the whole distribution.
4.Binomial Probability Formula★
$P (X = x) =^{n} C_{x} p^{x} q^{n - x}, x = 0, 1, \dots, n$
$n$ : Number of trials · $x$ : Number of successes (0 to n) · $p, q$ : Probabilities of success and failure in one trial
ⁿCₓ counts the arrangements of x successes among n trials; pˣqⁿ⁻ˣ is the probability of any one such arrangement.
5.Probabilities as Terms of a Binomial Expansion
$\sum_{x = 0}^{n}^{n} C_{x} p^{x} q^{n - x} = (q + p)^{n} = 1$
The probabilities P(X = 0), P(X = 1), …, P(X = n) are the successive terms of (q + p)ⁿ, which is why they add up to 1.
6.No Successes and All Successes
$P (X = 0) = q^{n}, P (X = n) = p^{n}$
The two end terms of the expansion. ⁿC₀ = ⁿCₙ = 1.
7.At Least One Success★
$P (X \geq 1) = 1 - P (X = 0) = 1 - q^{n}$
Use the complement. This also answers 'how many trials are needed' questions: solve 1 − qⁿ > k for the least whole number n.
8.Cumulative Probabilities
$P (X \leq r) = \sum_{x = 0}^{r} P (X = x), P (X \geq r) = 1 - P (X \leq r - 1)$
Choose whichever side has fewer terms. P(X > r) = 1 − P(X ≤ r), not 1 − P(X ≤ r − 1).
9.Fair Coin Case
$p = q = \frac{1}{2} \Rightarrow P (X = x) = \frac{^{n} C _{x}}{2 ^{n}}$
For tosses of a fair coin (or any p = 1/2 situation) every probability has denominator 2ⁿ.
10.Mean of a Binomial Distribution★
$E (X) = μ = n p$
The expected number of successes. Units are the same as X (number of successes).
11.Variance of a Binomial Distribution★
$Var (X) = σ^{2} = n pq$
Equivalently np(1 − p). It is largest, for a fixed n, when p = q = 1/2.
12.Standard Deviation
$σ = n pq$
Square root of the variance. Do not confuse σ with σ².
13.Variance is Less Than the Mean
$\frac{Var ( X )}{E ( X )} = \frac{n pq}{n p} = q < 1 \Rightarrow Var (X) < E (X)$
Holds whenever 0 < p < 1. If a question gives a variance larger than the mean, no binomial distribution fits.
14.Finding n and p from Mean and Variance
$q = \frac{n pq}{n p}, p = 1 - q, n = \frac{n p}{p}$
Divide the variance by the mean to get q. n must come out as a positive whole number — if it does not, recheck the arithmetic.