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The Binomial Probability Distribution and Related Topics

Foundational StatisticsLecture 9

Binomial probability distribution and its properties

This lecture and its associated materials have been produced by Dr. Wittaya Kanchanapusakit (PhD, Cambridge) and Dr. Phanida Saikhwan (PhD, Cambridge) of iAcademy for the purposes of lecturing on the above described subject and the material should be viewed in this context. The work does not constitute professional advice and no warranties are made regarding the information presented. The Author and iAcademy do not accept any liability for the consequences of any action taken as a result of the work or any recommendations made or inferred. Permission to use any of these materials must be first granted by iAcademy.

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Agenda• Review of week 8• Week 9 Lecture Material– Binomial probabilities– Additional properties of the binomial distribution

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Review of Week 8• The college student senate is sponsoring a spring break

Caribbean cruise raffle. The proceeds are to be donated to the Samaritan Centre for the Homeless. A local travel agency donated the cruise, valued at $2,000. The students sold 2852 raffle tickets at $5 per ticket.

• Kevin bought six tickets. What is the probability that Kevin will win the spring break cruise to the Caribbean? What is the probability that Kevin will not win the cruise?

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Review of Week 8(2)• Kevin bought six tickets. What is the probability

that Kevin will win the spring break cruise to the Caribbean? What is the probability that Kevin will not win the cruise?

• Let’s discuss!– What is a variable, x? What are possible values of x?

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Review of Week 8(3)• Let’s discuss!– Kelvin bought six tickets. What are possible

outcomes?

– What is probability that each ticket will win/lose?

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outcome ProbabilityLLLLLLWLLLLLLWLLLLLLWLLLLLLWLLLLLLWLLLLLLW

Review of Week 8(4)• Let’s discuss– What is the probability that Kevin will win the Spring

break Caribbean cruise?• Probability of each outcome is as follows:

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Review of Week 8(5)• Note:– We can also determine the number of outcomes

using the combination rule

– When calculating probability that Kevin will win we can multiply the no. of outcome that one ticket will win with probability that one ticket will win

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Review of Week 8(6)• Let’s discuss– What are Kevin’s expected earnings?

– How much did Kevin effectively contribute to the Samaritan Centre for Homeless?

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Any Questions?

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Binomial probabilities• Binomial probabilities are from

binomial experiment.• Binomial experiment or Bernoulli

experiment is when there are exactly two possible outcomes (for each trial) of interest. – e.g. tossing a coin where outcomes are

either head or tail

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Features of a binomial experiment• There are a fixed number of trials, n.• The n trials are independent and repeated under

identical conditions.• Each trial has only two outcomes: success (S) and

failure (F).• For each individual trial, the probability of success (p) is

the same. Since only two outcomes,– p+q = 1; q = probability of failure

• The central problem of a binomial experiment is to find the probability of r success out of n trials.

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Examples of binomial trials

• Watch this video. Play Video:– Bernoulli trials in everyday life

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Any Questions?

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Let’s try analysing binomial experiment• Blood type of 18 people

selected randomly from the population was tested.

• Given that 9% of the population has blood type B.

• What is the probability that three of these 18 people have blood type B?

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Let’s try analysing• Is this binomial experiment?– Yes, success = has type B blood and failure = does not

have type B blood• What are probabilities of success and failures?– 9% of the population has blood type B p = 0.09– q = 1 – p = 0.91

• How many trials?– n = 18

• We wish to compute the probability of 3 success out of 18 trials.– r = 3

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Computing probabilities for a binomial experiment(1)• Using formula for the binomial probability distribution

• Where – n = number of trials– p = probability of success on each trial– r = random variable representing the number of

successes out of n trials ()– ! = factorial notation (Recall 4! = 4 3 2 1)∙ ∙ ∙– Cn,r is called binomial coefficient

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Let’s try

• Continue with the blood B type example.• n = 18, r = 3, p = 0.09, q = 0.91

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Any Questions?

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Computing probabilities for a binomial experiment(2)• Using a binomial distribution table (Table 3 of statistical

tables given).• The table gives the probability of – r successes in n independent trials, – each with probability of success p.

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Computing probabilities for a binomial experiment(3)

• e.g. n = 6 and p = 0.50 find P(4) by looking at the entry– in the row headed by 4 and – the column headed by 0.50

• P(4) = 0.234

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Let’s try• A biologist is studying a new hybrid tomato. It is known

that the seeds of this hybrid tomato have probability 0.70 of germinating. The biologist plants six seeds.

• What is the probability that exactly four seeds will germinate?

• n = 6, r = 4, p (success = grow) = 0.70, q = 1-0.70 = 0.30

• Use binomial distribution table given, look for the section with n = 6, column headed by p = 0.70 and the row headed by r = 4.

• P(4) = 0.324

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Let’s try more …• What is the probability that at least four seeds will

germinate?

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Any Questions?

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Common expressions and corresponding inequalitiesExpression InequalityFour or more successesAt least four successesNo fewer than four successesNot less than four successes

That is, r = 4, 5, 6, …, n

Four or fewer successesAt most four successesNo more than four successesThe number of successes does not exceed four

That is, r = 0, 1, 2, 3 or 4

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Common expressions and corresponding inequalities(2)Expression InequalityMore than four successesThe number of successes exceeds four

That is, r = 5, 6, 7, …, n

Fewer than four successesThe number of successes is not as large as four

That is, r = 0, 1, 2, 3

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Any Questions?

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• A rarely performed and somewhat risky eye operation is known to be successful in restoring the eyesight of 30% of the patients who undergo the operation. A team of surgeons has developed a new technique for this operation that has been successful in four of six operations. Does it seem likely that the new technique is much better than the old?

• How do you tell whether the new technique is better than the old one?

Discussion: Find P(r)

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Discussion: Find P(r)• Find the probability of at least four successes in

six trials for the old technique. What are values of n, p, q and r?

• Use the formula: P(r) = Cn,r pr qn-r to find P(4)

• What is P(4) using the table?

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Discussion: Find P(r)• P(4) is not the answer as we are looking for the

probability of at least four success out of the six trials or P(r ≥ 4). What should we find then?

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Discussion: Find P(r)• Is the new technique better than the old?

• This means one of the following two things may happening:

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Any Questions?

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Graphing a binomial distribution• How to graph a binomial distribution– Place r values on the horizontal axis.– Place P(r) values on the vertical axis.– Construct a bar over each r value extending from

r - 0.5 to r + 0.5. The height of the corresponding bar is P(r)

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Let’s try graphing!• Jim enjoys playing basketball. He figures that he makes

about 50% of the field goals he attempts during a game.

• Make a histogram showing the probability that Jim will make 0, 1, 2, 3, 4, 5, or 6 shots out of six attempted field goals.

• This is a binomial experiment with – n = 6 trials and – p = 0.5

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r P(r)3456

Basketball problem• Find P(r) values for n = 6 and p = 0.5• Use the given table

r P(r)0 0.0161 0.0942 0.234

Your turn to complete the table!

0.3120.2340.0940.016

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0 1 2 3 4 5 60

0.050.1

0.150.2

0.250.3

0.35

r

P(r)

Basketball problem(2)• Use the values of P(r) to make a histogram

r P(r)3456

r P(r)0 0.0161 0.0942 0.234

0.3120.2340.0940.016

The height of bar over r = P(r) = area

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Basketball problem(3)• The graph is

symmetrical!• When p = 0.5, the

graph of the binomial distribution will be symmetrical no matter how many trials we have.

0 1 2 3 4 5 60

0.050.1

0.150.2

0.250.3

0.35

r

P(r)

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Any Questions?

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Mean and standard deviation• How to compute m and s for a binomial

distribution• Expected number of successes for the random variable

r : • Standard deviation for the random variable r: • Where – r is a random variable representing the number of

successes in a binomial distribution– n is the no. of trials– p is the probability of success on a single trial– q = 1- p is the probability of failure on a single trial

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• What is the expected number of goals Jim will make?• What is the standard deviation of the binomial

distribution of the number of successful field goals Jim makes?

Basketball problem(4)

• Recall: n = 6, p = 0.5, q = 0.5• Expected no. of goals = = m np = 6(0.5) = 3• Standard deviation, s =

The mean is not only the balance point of the distribution but also the expected value of r.

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What is expected from binomial trials? What is standard deviation?

• Watch this video. Play Video:– Expected value and standard deviation of a binomial

distribution

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Any Questions?

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Unusual values• For a binomial distribution, it is unusual for the

number of successes r to be – Higher than – Lower than

• e.g. consider a binomial experiment with 20 trials for which p = 0.70. The expected number of successes is 14, with a standard deviation of 2.– A number above 19 or below 9 would be considered

unusual but possible.

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Any Questions?

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Summary• A binomial experiment consists of a fixed

number n of independent trials repeated under identical conditions.– Two outcomes for each trial: success and failure.– The probability p of success on each trial is the

same.• The number of successes r in a binomial

experiment is the random variable for the binomial probability distribution.– The probabilities can be computed using a formula

or using the given table or calculator.

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Summary(2)• For a binomial distribution,

– Where q = 1 - p• For a binomial experiment, the number of

successes outside the range of to is unusual but can occur!

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Tutorial• Bring the given table (binomial probabilities) to

the tutorial.• Also bring the table to Lecture 10, you will need

it for revision.

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