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Question

Which of the following situations can be modelled by a binomial distribution? Explain Please for each.

a) A child rolls a die ten times and counts the number of 3's.

b) The first player in a free-throw basketball competition has a free-throw success rate of 88.4%. A second player takes over when the first player misses the basket.

c) A farmer gives 12 of the 200 cattle in a herd an antibiotic. The farmer then selects 10 cattle at random to test for infections to see if the antibiotic was effective.

d) A factory producing electric motors has a 0.2% defect rate. A quality control inspector needs to determine the expected number of motors that would fail in a day's production.

Answer 1

A binomial distribution is characterized by independent trials,

each having a binary outcome, with common probability.

a)

Rolling a die are independent trials, each yields a 3 or not,

all with the probability of 1/6.

The probability of getting k 3's after 10 trials follows a binomial distribution.

b)

Each throw is an independent trial (ignoring psychology),

and each throw either succeeds of not, (assuming) with the same probability.

The number of successful throws follows binomial distribution.

We are told that the mean of the distribution is 88.4%.

I do not know what you are expected to say about the second player.

c)

- The number of cattle that get sick would follow a binomial distribution

if one getting sick were independent of others getting sick.

I do not know how realistic an assumption is that.

- Selection of cattle for vaccination or testing is not a binomial process,

because it is selection WITHOUT replacement.

- Determining which cattle selected for testing is sick constitutes independent trials.

If all those trials had the same probability then the vaccine is not effective.

In other words, the determination of vaccine effectiveness amounts to determining

whether disease among selected cattle follows a binomial distribution.

d)

- It is known that one electric motor being defective is not independent

of others being defective.

However, in the absence of information about correlations, quality engineers

often assume that failures are independent.

If so, the number of defective motors would follow a binomial distribution.

- Selecting motors for testing is not a binomial process,

because it is selection WITHOUT replacement.

- In selecting motors for test, the inspector would try to chose them

randomly, so that one failing is independent of others.

In other words, the goal of the selection process is to make the number

of failing motors follow a binomial distribution.