Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- A coin lands heads 12 times in 30 tosses. What is the relative frequency of heads, as a decimal?
Answer
0.4. 12/30 = 4/10. - Find 15% of 200.
Answer
30. 10% is 20 and 5% is 10. - Write 0.35 as a fraction in its simplest form.
Answer
7/20. 35/100 divided by 5 on top and bottom.
Words used in this lesson
- expected frequency
- the number of times to expect an event, probability times trials. For example: 30 sixes in 180 rolls.
- simulation
- using random numbers to act out a real event many times. For example: digits standing for children.
- trial
- one go at the experiment or simulation. For example: one simulated family.
- random
- each outcome equally likely, with no pattern. For example: a digit chosen from 0 to 9.
- estimate
- a value found from an experiment, close to but not exactly the truth. For example: 0.845 for 7/8.
Expect, then simulate
A family has 3 children, and each is as likely to be a girl as a boy. How likely is it that at least one is a girl? Before you open the figure, write your estimate as a fraction.
You could list all the outcomes. Or you could simulate: let random numbers stand in for the children, make many pretend families, and count. The figure plays a fixed run of pretend families, the same every time. Press “Show the theory” only after you have read the simulation.
Expected frequency = probability × number of trials. It is the number of times to expect on average, so the real count is usually close to it but need not be equal.
A simulation stands in for the real event. Choose random digits so that the share of digits for “yes” equals the probability, run many trials, and use the relative frequency as the estimate.
Slide the families from 10 up to 200, read the relative frequency each time, then press the button and compare.
10 simulated families, 7 with at least one girl: the relative frequency is 7/10 = 0.700. Press Show the theory to compare.
Show me the steps
- 10 families: 7 have at least one girl. Relative frequency 7/10 = 0.700.
- 50 families: 43 have at least one girl, 0.860.
- 100 families: 86 have at least one girl, 0.860.
- 200 families: 169 have at least one girl, 0.845. Now press Show the theory and compare.
Expected does not mean certain. After you press the button, compare the expected number with the simulated count: being a few away is not a mistake. An expected frequency can even be a number that cannot happen, such as a fraction of a family.
Worked examples
A fair die is rolled 300 times. How many sixes should you expect?
- P(six) = 1/6.
- Expected frequency = 1/6 × 300 = 50 sixes.
A player scores 70% of her free throws. Describe a simulation of one throw using the digits 0 to 9.
- Ten digits are equally likely, and 70% of 10 digits is 7 digits.
- Let the digits 0, 1, 2, 3, 4, 5, 6 mean “scores” and 7, 8, 9 mean “misses”.
- Each random digit is one throw. Count the scores in many digits, and divide by how many digits you used.
Your go. The first step is given. A bakery sells out of bread on 30% of days. To simulate one day with a random two-digit number from 00 to 99, how many of those 100 numbers should stand for “sells out”?
- There are 100 equally likely numbers: 00, 01, …, 99.
- Finish it, then check.
Check the answer
30% of 100 = 30 numbers, for example 00 to 29.
Check before you go on
1. A fair die is rolled 120 times. How many sixes would you expect?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A seed germinates with probability 0.9. Of 200 seeds, how many would you expect to germinate?
3. A spinner has 5 equal sections and 2 are blue. It is spun 150 times. How many blues would you expect?
Level 2 · Building
4. In a survey, 48 of 200 people said they cycle to work. Use this to predict how many of 1500 people cycle to work.
5. A fair coin is tossed 200 times and lands heads 109 times. How many more heads is that than the expected number?
6. Why might someone use random digits to simulate an event instead of working out the probability by theory?
Level 3 · Stretch
7. In a game you win if a fair die shows 5 or 6. You play 90 times. How many wins would you expect?
8. An event has probability 0.15. In some number of trials you would expect it to happen 30 times. How many trials is that?
Something to think about
The simulation gives 0.845 after 200 families and the theory says 0.875. A friend says the simulation must be wrong. Another says the theory must be wrong. What would you say to each, and what would you do to settle it?
One way to think about it
Neither is wrong. A simulation is an estimate, and 200 families can be a little off. Theory counts every outcome exactly. To settle it, run more families and see whether the relative frequency settles near 0.875, and check the theory by listing all 8 outcomes again.
Exit ticket
9. Rashida simulates an event with probability 40% using one random digit from 0 to 9. Which choice is correct?