Binomial Distribution (AA SL)
The binomial distribution models the number of "successes" in a fixed number of independent trials, each with the same probability of success - repeated coin flips, quiz guesses, defective items on a production line. This page covers when \(X\sim B(n,p)\) applies and how to compute its probabilities on the GDC, with worked examples and the mistakes that lose marks. It's part of the broader Random Variables & Distributions topic.
17 questions on this sub-topic.
Setup and formula
Covered under IB syllabus reference SL4.8: the binomial distribution, including its mean and variance. In examinations, binomial probabilities should be found using technology.
Notation and mean
\(X\sim B(n,p)\), \(E(X)=np\)
\(n\) is the number of trials and \(p\) is the probability of success on each one. This mean formula is in the formula booklet.
Conditions for binomial
Fixed number of trials \(n\); two outcomes per trial; constant probability \(p\); independent trials.
All four conditions must hold before you can write \(X\sim B(n,p)\) - not in the formula booklet, but it's the key exam idea examiners test.
Need the full syllabus wording and formula-booklet reference table? See Random Variables & Distributions.
Worked examples
\(X \sim B(10, 0.3)\).
Find \(P(X = 4)\), to 3 significant figures.
Worked solution
\(P(X=4)=\binom{10}{4}(0.3)^4(0.7)^6.\) M1
\(\approx 0.200.\) A1
On the GDC: \(\texttt{binompdf(10, 0.3, 4)}\approx 0.200.\)
A multiple-choice quiz has 12 questions, each with probability 0.25 of a correct guess.
Find \(P(\text{at least one correct})\) when guessing all.
Worked solution
At least one means not zero: \(P(X\ge 1)=1-P(X=0).\) M1
\(P(X=0)=(0.75)^{12}\approx 0.0317.\) A1
\(P(X\ge1)=1-0.0317=0.9683\approx0.968\) (3 s.f.). A1
On the GDC: \(1-\texttt{binompdf(12, 0.25, 0)}\approx 0.968.\)
\(X\sim B(8, 0.25).\) 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
\(X\sim B(20, 0.4).\)
(a) Find the mean.
(b) Find the variance.
Worked solution
(a) \(E(X) = np = 20(0.4)\) M1
\(= 8.\) A1
(b) \(\text{Var} = np(1-p) = 20(0.4)(0.6)\) M1
\(= 4.8.\) A1
Common mistakes
- Not checking the four binomial conditions. A situation is only binomial if trials are fixed in number, independent, and have a constant probability of success - sampling without replacement from a small population breaks this.
- Using binompdf when a range is needed. \(P(X\ge 2)\) or \(P(X\le 5)\) call for the cumulative function binomcdf (with a complement where needed), not binompdf, which only gives one exact value.
- Confusing \(n\) and \(p\) in the GDC entry. The number of trials always goes first and the probability second - swapping them silently produces a completely different (and wrong) distribution.
Ready to practise properly?
17 binomial distribution questions, marked instantly like the real exam.
Quick answers
What are the conditions for a binomial distribution?
A fixed number of trials \(n\), exactly two outcomes per trial, a constant probability of success \(p\), and independent trials. All four must hold before you can write \(X\sim B(n,p)\).
How do you calculate binomial probabilities on the GDC?
Use \(\texttt{binompdf(n, p, x)}\) for an exact value \(P(X=x)\), or \(\texttt{binomcdf(n, p, x)}\) for a cumulative probability \(P(X\le x)\). IB exams expect these to be found using technology.