Binomial Distribution and Expectation
For n independent trials with success probability p, P(X = r) = nCr pr (1-p)^(n-r), with mean np and variance np(1-p).
What is Binomial Distribution and Expectation?
For n independent trials with success probability p, P(X = r) = nCr pr (1-p)^(n-r), with mean np and variance np(1-p).
Key formula / rule: All four binomial conditions must hold before the formula applies.
Key points
- Evaluate whether a situation satisfies the binomial conditions.
- Compute binomial probabilities, mean and variance.
Common exam trap
Applying the binomial formula when trials are drawn without replacement.
Definitions
- Term
Binomial Distribution and Expectation
- Meaning
For n independent trials with success probability p, P(X = r) = nCr pr (1-p)^(n-r), with mean np and variance np(1-p).
- Term
Binomial Distribution and Expectation — explanation
- Meaning
The binomial model requires fixed n, two outcomes, constant p and independence; failing any of these invalidates the formula. Mean and variance follow directly from the per-trial values.
Learning objectives
Evaluate whether a situation satisfies the binomial conditions.
Compute binomial probabilities, mean and variance.
Formulae
- Key point
All four binomial conditions must hold before the formula applies.
- Key point
Mean is np and variance is np(1-p), so variance is always less than the mean.
- Key point
The distribution is symmetric only when p = 0.5.
Prerequisites
MAT-U5-PS-T1-S1-C2
Common mistakes
Applying the binomial formula when trials are drawn without replacement.
Using np(1-p) as the standard deviation instead of the variance.
Keywords
Binomial
Distribution
Expectation
Practice preview
A fair coin is tossed 6 times. What is the probability of getting exactly 4 heads?…
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For a Binomial distribution with 8 trials and probability of success 0.25, what is its variance?…
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A random variable X follows a Binomial distribution such that its mean is 6 and its variance is 4.2. What are the values of n and p?…
medium