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Binomial conditions, the distribution table and graph shape

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Five independent trials each have success probability two fifths. Which expression gives the probability of exactly two successes?

Year 12, VCE Units 3 & 4, Chapter 11, 4 difficulty levels, No calculatorBuild a worksheet

The binomial distribution table

A binomial variable counts successes in a fixed number of independent trials with the same success probability. Each row of its table gives the probability of one possible success count

\[\Pr(X=x)=\binom{n}{x}(1-p)^{n-x}p^x\]
  1. Check the conditions Use a binomial model only for a fixed number of independent trials with two outcomes and a constant success probability \(\displaystyle X\sim\operatorname{Bi}\left(5,\frac{2}{5}\right)\)
  2. Use the binomial term For the required success count, include the combination coefficient, the failure power and the success power \(\displaystyle \Pr(X=2)=\binom{5}{2}\left(\frac{3}{5}\right)^3\left(\frac{2}{5}\right)^2\)
  3. Complete the table List every success count from zero to the full trial count. The probabilities in a complete table add to one \(\displaystyle x=0,1,2,3,4,5,\quad \sum_{x=0}^{5}\Pr(X=x)=1\)
  4. Relate success probability to shape For a positive number of trials, with both success and failure possible, a success probability below one half gives positive skew, one half gives symmetry, and above one half gives negative skew \(\displaystyle p<\frac{1}{2}\Rightarrow\text{positive skew},\quad p=\frac{1}{2}\Rightarrow\text{symmetric}\)

Worked example

A binomial variable counts successes in four independent trials, each with success probability three fifths. Find the probability of exactly one success

\[X\sim\operatorname{Bi}\left(4,\frac{3}{5}\right)\]
  1. \(\displaystyle 1-p=1-\frac{3}{5}=\frac{2}{5}\)Find the failure probability from the complement of the success probability
  2. \(\displaystyle \binom{4}{1}=4\)There are four possible positions for the one success
  3. \(\displaystyle \Pr(X=1)=\binom{4}{1}\left(\frac{2}{5}\right)^3\left(\frac{3}{5}\right)\)Use three failures and one success in the binomial formula
  4. \(\displaystyle \Pr(X=1)=4\times\frac{8}{125}\times\frac{3}{5}=\frac{96}{625}=0.1536\)Multiply the coefficient and the two probability powers

Common mistakes

Omitting the combination coefficient
✕\(\displaystyle \Pr(X=2)=\left(\frac{3}{5}\right)^3\left(\frac{2}{5}\right)^2=\frac{108}{3125}\)✓\(\displaystyle \Pr(X=2)=\binom{5}{2}\left(\frac{3}{5}\right)^3\left(\frac{2}{5}\right)^2=\frac{216}{625}\)
The coefficient counts the different positions for the two successes, so it must multiply the powers
Using non-complementary exponents
✕\(\displaystyle \Pr(X=2)=\binom{5}{2}\left(\frac{3}{5}\right)^2\left(\frac{2}{5}\right)^2\)✓\(\displaystyle \Pr(X=2)=\binom{5}{2}\left(\frac{3}{5}\right)^3\left(\frac{2}{5}\right)^2\)
The failure and success exponents must add to the total number of trials
Leaving out table endpoints
✕\(\displaystyle x\in\{1,2,3,4\}\)✓\(\displaystyle x\in\{0,1,2,3,4,5\}\)
A binomial variable can have zero successes or all successes, so both endpoints belong in the table

Difficulty levels

introRecognise a binomial situation and complete a small distribution table
standardConstruct a full table for more trials and read a requested probability
challengeInterpret the graph shape or distinguish a specified order from an unordered success count
examCombine a binomial verdict or formula choice with a table probability consequence

Questions about this topic

Why is the combination coefficient needed?+

The coefficient counts the different orders in which the required number of successes can occur among the trials

What success counts belong in a binomial table?+

Every whole-number count from zero to the full trial count belongs, including no successes and success on every trial

When is a binomial distribution symmetric?+

For a positive number of trials, with both success and failure possible, the distribution is symmetric when the success probability is one half. A probability below one half gives positive skew, and a probability above one half gives negative skew

Can I use a calculator?+

No. This topic is practised without a 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