Probability of One Event
Estimating Probabilities Using Relative Frequency

Sometimes it is possible to calculate values for the probability of an event by symmetry arguments, like tossing a coin and getting a head. For other events probabilities can be estimated by using results of experiments or records of past events.

Worked Examples

1

In a town in Yorkshire there was rain on 18 days in February 2009. Use this information to estimate the probability that it rained there on a particular day in February 2009.

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It rained on 18 out of the 28 days, so the relative frequency of rain is

=

So the probability that it rained on any day in February can be estimated as .
2

Henry carries out an experiment with a piece of buttered toast. He drops it 50 times and finds that 35 times it lands butter side down. Use these results to estimate the probability that a piece of toast lands butter side up when dropped.

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The toast landed butter side down 35 of the 50 times, so the relative frequency is = .

So the probability that the toast lands butter side down can be estimated as . Hence, the probability of it landing butter side up is estimated as .

Exercises

Mrs Oliver observed that her bus was late on 6 of her 24 work days in March.

Estimate the probability that the bus was late on any one day.

A football team plays on average 40 matches each season and wins 32 of them.

Estimate the probability that this team wins a match.

Six students play regularly in a chess club. The number of games that each student has won is recorded in the table below.

PlayerGames WonGames Lost
Tom410
Andrew73
Danielle39
Rachel416
Connor612
Malene126
(a)

Use this data to find the probability that each student wins a match.

Tom

Andrew

Danielle

Rachel

Connor

Malene

(b)

Which student is the best player?

(c)

Which student is the worst player?

(d)

If Connor played Tom, who do you think would be more likely to win?

A garage records the number of cars that they sell each week over a 24 week period. The numbers for each week are given below.

3, 4, 8, 6, 5, 7, 4, 3, 6, 5, 2, 4,

5, 7, 6, 9, 2, 4, 5, 6, 7, 4, 3, 5.

Use this data to establish the probability that in any week;

(a)
more than 5 cars are sold,
(b)
fewer than 5 cars are sold,
(c)
exactly 5 cars are sold.

A gardener plants 40 seeds and 32 of them produce healthy plants.

(a)

Estimate the probability that a seed produces a healthy plant.

(b)

If 120 seeds were planted, how many healthy plants can the gardener expect to obtain?

Investigation

A student types 3 different letters and 3 different addresses on 3 envelopes. She puts the letters into the envelopes randomly and sends then to 3 of her friends, A, B and C. What is the probability that

(a) only one of her friends will receive the correct letter,

(b) only two of them will receive the correct letters?