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Exact probability of repeated independent successes

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Which expression represents exactly 3 successes in 7 independent trials when the success probability is p?

Year 12, VCE Units 3 & 4, Chapter 11, 2 difficulty levels, Calculator allowedBuild a worksheet

Exact binomial probability

A binomial variable counts successes in a fixed number of independent trials with the same success probability. For an exact success count, count the possible positions of the successes and include the complementary failures

\[\Pr(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\]
  1. Use the combination coefficient The coefficient counts which trial positions contain the successes \(\displaystyle \binom{6}{2}=15\)
  2. Use the complementary probability The failure probability complements the success probability \(\displaystyle p=\frac{2}{5},\quad 1-p=\frac{3}{5}\)
  3. Make the exponents total n The success and failure exponents together count every trial \(\displaystyle 2 + (6-2) = 6\)
  4. Add disjoint success counts For a cumulative event, add the exact terms for the allowed success counts, or use a complement when it is shorter \(\displaystyle \Pr(X\ge 2)=\Pr(X=2)+\Pr(X=3)+\cdots\)

Worked example

A repeated independent trial has success probability one third and is performed 6 times. Find the probability of exactly 2 successes

\[\Pr(X=2)=\binom{6}{2}\left(\frac{1}{3}\right)^2\left(\frac{2}{3}\right)^4\]
  1. \(\displaystyle p=\frac{1}{3},\quad 1-p=\frac{2}{3},\quad n=6,\quad r=2\)Identify the success probability, failure probability, trial count and success count
  2. \(\displaystyle \binom{6}{2}=\frac{6\times 5}{2\times 1}=15\)Count the positions that can contain the 2 successes
  3. \(\displaystyle \Pr(X=2)=15\times\left(\frac{1}{3}\right)^2\times\left(\frac{2}{3}\right)^4\)Use the binomial term with 2 successes and 4 failures
  4. \(\displaystyle 15\times\frac{1}{9}\times\frac{16}{81}=\frac{240}{729}=\frac{80}{243}\)Multiply the exact factors and reduce the fraction

Common mistakes

Omitting the combination coefficient
✕\(\displaystyle \Pr(X=2)=\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^3=\frac{1}{32}\)✓\(\displaystyle \Pr(X=2)=\binom{5}{2}\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^3=\frac{5}{16}\)
The combination coefficient counts the possible positions of the successes, so it cannot be omitted
Using equal success and failure exponents
✕\(\displaystyle \binom{5}{2}\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^2=\frac{5}{8}\)✓\(\displaystyle \binom{5}{2}\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^3=\frac{5}{16}\)
The failure exponent is the remaining number of trials, so the two exponents must add to 5
Multiplying cumulative terms
✕\(\displaystyle \Pr(X\ge 2)=\Pr(X=2)\times\Pr(X=3)\times\Pr(X=4)\times\Pr(X=5)\)✓\(\displaystyle \Pr(X\ge 2)=\Pr(X=2)+\Pr(X=3)+\Pr(X=4)+\Pr(X=5)\)
Different success counts are disjoint alternatives, so their probabilities are added

Difficulty levels

challengeCumulative or range probabilities with a direct or complement strategy
examSelect the correct term set and formula data, then calculate one exact result

Questions about this topic

Why is the combination coefficient needed?+

A success count does not specify which trials were successful. The combination coefficient counts all possible positions of those successes

Why is the failure exponent the trial count minus the success count?+

There are n total trials and r successes, so the remaining trials are failures

When is a complement useful for a cumulative probability?+

Use a complement when the success counts outside the requested event form a shorter set of terms. Subtract their exact total from 1

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 11: Binomial distribution

In this chapter

Bernoulli trials and the Bernoulli distribution11A Bernoulli trials and the Bernoulli distribution: Finite events, probability assignments and complementsSolve a quadratic exactly with the quadratic formula11A Bernoulli trials and the Bernoulli distribution: Round to decimal places and significant figures11B Binomial distribution and parameters: Binomial conditions, the distribution table and graph shapeMean, variance, standard deviation and parameter recovery of a binomial variableExact probability of repeated independent successesEvaluate factorial and combination expressionsSolve a linear equation11B Binomial distribution and parameters: Round to decimal places and significant figuresCumulative, at-least-one and conditional probabilities of a binomial variableSmallest trial count for an at-least-one threshold11C Binomial distribution events: Finite events, probability assignments and complementsConditional probability from exact representationsSolve and interpret a linear inequation11D Binomial shape and specified order: Binomial conditions, the distribution table and graph shapeProbabilities of sequences of independent eventsDescribe the shape of a distribution