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Smallest trial count for an at-least-one threshold

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Binomial model
\(\displaystyle X\sim\operatorname{Bi}\left(n,\frac{1}{4}\right)\)

Which expression gives the probability of zero successes?

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

The smallest trial count

The probability of at least one success is found from the complement of zero successes. To find the smallest trial count, solve the resulting inequality and then choose the first integer that satisfies it

\[\Pr(X\ge 1)=1-(1-p)^n\]
  1. Find zero successes Zero successes means failure on every one of the n trials \(\displaystyle \Pr(X=0)=(1-p)^n\)
  2. Use the complement At least one success is the complement of zero successes \(\displaystyle \Pr(X\ge 1)=1-\Pr(X=0)=1-(1-p)^n\)
  3. Set the threshold Use a strict inequality when the probability must exceed the stated threshold t. \(\displaystyle 1-(1-p)^n>t\)
  4. Choose the first integer If the logarithmic bound is not an integer, round up and check the adjacent integer below it \(\displaystyle n>3.71\Longrightarrow n=4\)

Worked example

An independent trial has success probability two fifths. Find the smallest trial count for which the probability of at least one success is greater than 0.85

\[\Pr(X\ge 1)=1-\left(\frac{3}{5}\right)^n>0.85\]
  1. \(\displaystyle 1-p=1-\frac{2}{5}=\frac{3}{5}\)Find the failure probability from the complement of p.
  2. \(\displaystyle \Pr(X\ge 1)=1-\left(\frac{3}{5}\right)^n>0.85\)Use the complement of zero successes for the at-least-one event
  3. \(\displaystyle \left(\frac{3}{5}\right)^n<0.15\)Rearrange the strict threshold inequality to isolate the probability of zero successes
  4. \(\displaystyle n=3:\quad 1-\left(\frac{3}{5}\right)^3=1-\frac{27}{125}=\frac{98}{125}\)Substitute three trials into the at-least-one probability
  5. \(\displaystyle \frac{98}{125}=0.784<0.85\)Three trials do not meet the threshold
  6. \(\displaystyle n=4:\quad 1-\left(\frac{3}{5}\right)^4=1-\frac{81}{625}=\frac{544}{625}\)Increase the trial count to four and calculate again
  7. \(\displaystyle \frac{544}{625}=0.8704>0.85\)Four trials meet the threshold. This probability increases with the trial count, so four is the smallest value

Common mistakes

Using np for an at-least-one threshold
✕\(\displaystyle n\times\frac{1}{4}>0.600\)✓\(\displaystyle 1-\left(\frac{3}{4}\right)^n>0.600\)
The at-least-one probability is not np; it is one minus the probability of no successes
Rounding the logarithmic bound down
✕\(\displaystyle n=\left\lfloor\frac{\log(0.15)}{\log(0.6)}\right\rfloor=3\)✓\(\displaystyle n=\left\lceil\frac{\log(0.15)}{\log(0.6)}\right\rceil=4\)
The smallest whole number above a non-integer lower bound is found by rounding up, not down
Using the success probability for zero successes
✕\(\displaystyle \Pr(X\ge 1)=\left(\frac{2}{5}\right)^n\)✓\(\displaystyle \Pr(X\ge 1)=1-\left(\frac{3}{5}\right)^n\)
Zero successes uses one minus the success probability, and at least one success is its complement

Difficulty levels

introUse a supplied complement relation with a small trial count
standardForm the complement and solve the threshold inequality
challengeUse a threshold near an exact boundary where integer rounding determines the answer
examChoose among solution strategies and report the smallest integer trial count

Questions about this topic

Why is the complement used for at least one success?+

At least one success and zero successes are complementary events, so subtract the zero-success probability from one to find the at-least-one probability

Why is the logarithmic answer rounded up?+

The inequality gives a lower bound for n, so the first whole number above a non-integer bound is required

How can the smallest trial count be checked?+

Calculate the probability for the candidate count and for one fewer trial; the candidate must exceed the threshold while the smaller count must not

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