Binomial Distribution (AI SL)
The binomial distribution \(X\sim B(n,p)\) models the number of successes in \(n\) independent trials, each with the same success probability \(p\) - defective items on a production line, correct guesses, sixes rolled. You'll find its probabilities on your GDC and its mean and variance from two short formulas. It's part of the broader Distributions topic.
28 questions on this sub-topic.
Mean, variance and probability
Covered under IB syllabus reference SL4.8: the binomial distribution, and its mean and variance. A formal proof of the mean and variance is not required, and in examinations binomial probabilities are found using technology rather than by hand.
Mean and variance
\(E(X)=np \qquad \text{Var}(X)=np(1-p)\)
Both are in the formula booklet. You only need \(n\) and \(p\) - no need to list out the whole distribution first.
Binomial probability
\(P(X=r),\ P(X\le r)\)
Found directly from \(n\), \(p\) and \(r\) using your GDC's binomial pdf (exactly) or cdf (at most) function - not calculated by hand.
Not in the formula booklet - GDC requiredNeed the full syllabus wording and formula-booklet reference table? See Distributions.
Worked examples
\(X\sim B(40, 0.1)\). Find the mean \(E(X)\).
Worked solution
\(X\sim B(40,0.1):\) \(n=40,\ p=0.1.\)
\(E(X)=np=40(0.1)\) A1 \(=4.\) A1 (No GDC needed - the mean of a binomial is simply \(np.\))
\(X\sim B(8, 0.25)\). Find \(P(X\ge2)\).
Worked solution
\(P(X\ge2) = 1 - P(0) - P(1).\) M1
\(P(0) = 0.75^8 \approx 0.1001\); A1 \(P(1) = 8(0.25)(0.75)^7 \approx 0.2670.\) A1
\(P(X\ge2) \approx 1 - 0.367\) M1 \(= 0.633.\) A1
\(X\sim B(8, 0.25).\) Find \(P(X\ge2).\)
Worked solution
\(P(X\ge2) = 1 - P(0) - P(1).\) M1
\(P(0) = 0.75^8 \approx 0.1001\); A1 \(P(1) = 8(0.25)(0.75)^7 \approx 0.2670.\) A1
\(P(X\ge2) \approx 1 - 0.367\) M1 \(= 0.633.\) A1
\(X\sim B(12, 0.5)\). Using a GDC:
(a) Find \(P(X\le 4)\).
(b) Find \(P(X\ge 8)\), to 3 significant figures.
Worked solution
(a) \(P(X\le 4)\). Cumulative up to 4: \(P(X\le 4)\) M1
\(\approx 0.194.\) A1
(b) \(P(X\ge 8)\). Use the complement: \(P(X\ge 8)=1-P(X\le 7)\approx 1-0.806\) M1
\(=0.194.\) A1
Common mistakes
- Confusing pdf and cdf. Binomial pdf gives \(P(X=r)\), "exactly \(r\)"; binomial cdf gives \(P(X\le r)\), "at most \(r\)". Reaching for the wrong one is the single most common lost mark on this topic.
- Forgetting "at least" needs a complement. \(P(X\ge r)\) is not directly on the GDC menu - compute it as \(1 - P(X\le r-1)\), being careful with the boundary term.
- Using the normal distribution for a discrete count. A count of successes from a fixed number of trials (e.g. number of heads in 20 flips) is binomial, not normal - check whether the data is discrete or continuous before choosing a model.
Ready to practise properly?
28 binomial-distribution questions, marked instantly like the real exam.
Quick answers
What is the formula for the mean of a binomial distribution?
If \(X\sim B(n,p)\), the mean is \(E(X)=np\) and the variance is \(\text{Var}(X)=np(1-p)\).
When should I use binomial pdf versus binomial cdf?
Use pdf for "exactly \(r\)" successes. Use cdf for "at most \(r\)" successes. For "at least" or "more than", find the complement using \(1-\)cdf. See the GDC guidance on the full Distributions page.