Expectation of Binomial Distribution
1. Research has shown that 1 in every 4 people is allergic to a certain type of pollen. A new
treatment is devised and is given to a group 20 student volunteers.
(a) What is the expectation of the number of allergic people in the group?
(b) What is the probability that
(i)
exactly two
(ii)
no more than two of the group are allergic?
(c) How large a sample would be needed for the probability of it containing at least one
allergic person to be greater than 99.9%?
(d) What assumptions have you made in your answer?
2. An unbiased die is rolled three times.
(a) What is the expected number of sixes?
(b) What is the most likely number of sixes?
3. Of the articles from a certain production line, 10% are defective. If a sample of 25 articles is
taken, find the expected number of defective articles.
4. On average, 20% of the bolts produced by a machine in a factory are faulty. Samples of 10
bolts are to be selected at random each day.
(a) Calculate, to 2 sf, the probability that, in any one sample, two bolts or less will be faulty.
(b) Find the expected value of the number of bolts in a sample which will not be faulty.
5. An unbiased die is rolled 4 times.
a. Calculate the probability distribution of the number of sixes.
b. Illustrate this distribution on a vertical line chart (because it is discrete).
c. What is the probability that the number of sixes is greater than the expectation?
6. How many dice must be rolled for the probability that no fives or sixes are showing to be
less than 1%?
7. A weighted coin, with P  head   0.46 , is tossed five times.
(c) What is the expected number of heads?
(d) What is the most likely number of heads?