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Binomial Distribution Mean & Variance

Probability · Class XII

n trials, success probability p. See mean np, variance np(1−p) and the chance of exactly k successes.

10
type a value
150
0.5
type a value
0.010.99
5
type a value
050
Live values
  • Mean5
  • Variance2.5
  • P(X = k)0.2461
xy
  • Variance n*p*(1-p) vs p

Formulas in this lab

  • Mean
    μ=np\mu = n p
  • Variance
    σ2=np(1−p)\sigma^2 = n p (1-p)
  • P(X = k)
    (nk)pk(1−p)n−k\binom{n}{k} p^k (1-p)^{n-k}
Tip: Variance is a downward parabola in p, peaking at p = 0.5 with value n/4.

Frequently asked questions

▶What is the binomial distribution?

If a trial succeeds with probability $p$ and you run $n$ independent trials, the probability of exactly $k$ successes is $P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$. The mean is $np$ and variance is $np(1-p)$.

▶How do I use the binomial distribution lab?

Slide $n$, $p$ and $k$. The lab returns $P(X = k)$, the mean $np$ and variance $np(1-p)$. Try $n = 10$, $p = 0.5$, $k = 5$: probability $\approx 0.246$, mean $5$, variance $2.5$. The probability peaks near $k = np$.

▶Where does the binomial show up in real life?

Quality control: probability that exactly $3$ out of $20$ Maruti car parts are defective if defect rate is $5\%$. Cricket: probability of hitting exactly $2$ sixes in $6$ balls. Genetics: number of dominant traits in offspring. The binomial is the workhorse of discrete probability.

▶Why use binomial versus normal distribution?

Binomial is exact for discrete success counts. When $n$ is large and $p$ isn't extreme, the binomial approximates the normal distribution with mean $np$ and standard deviation $\sqrt{np(1-p)}$. JEE often asks you to spot when to switch — typically when $np > 5$ and $n(1-p) > 5$.