Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- A fair 10-sided die is numbered 1 to 10. What is the probability of a prime number?
Answer
4/10 = 2/5. The primes are 2, 3, 5 and 7. - P(A) = 0.6. What is P(not A)?
Answer
0.4. 1 − 0.6. - A frequency table shows frequencies 5, 8, 12 and 5. What is the total frequency?
Answer
30. 5 + 8 + 12 + 5.
Words used in this lesson
- experimental probability
- how often something happened out of the trials. For example: 7 heads in 20 tosses is 7/20.
- relative frequency
- another name for experimental probability. For example: 0.35.
- trial
- one go at the experiment. For example: one roll of the die.
- theoretical probability
- the probability worked out from equally likely outcomes. For example: 1/6 for a six.
- fair
- with no outcome favoured over any other. For example: a fair die.
What happened, out of how many
A die is rolled 5 times and shows no sixes. Zoe says, “A six must be due.” Mo says, “This die must be unfair.” Before you open the figure, decide which of them, if either, has good evidence.
The experimental probability of an event is how often it happened out of the number of trials. It is also called the relative frequency. The figure plays back one fixed run of 500 rolls, the same every time you open it, and draws the relative frequency after each roll.
Experimental probability = number of times it happened ÷ number of trials. It estimates the probability, and it changes as more trials are added.
More trials tend to settle near the theory. A short run can be far from it. A long run is usually close, but it is not guaranteed to match exactly.
Slide from 5 rolls upward. Say where you think the line will be at 100 rolls before you look, and whether it will touch the dashed line.
10 rolls with 1 six: the relative frequency is 1/10 = 0.1, against a theory of 1/6 ≈ 0.167.
Show me the steps
- 20 rolls: 7 sixes, 7/20 = 0.35. More than twice the theory of 1/6 ≈ 0.167.
- 50 rolls: 12 sixes, 12/50 = 0.24.
- 100 rolls: 15 sixes, 15/100 = 0.15.
- 500 rolls: 75 sixes, 75/500 = 0.15. Close to 1/6 ≈ 0.167, and not equal to it.
A result is never “due”. A fair die has no memory, so the next roll is a six with probability 1/6 whatever happened before. And no sixes in 5 rolls is not strong evidence of anything: a fair die does that about two times in five (40%).
Worked examples
A drawing pin is dropped 50 times and lands point up 18 times. Find the experimental probability that it lands point up.
- It happened 18 times out of 50 trials.
- 18/50 = 0.36, which is 36%.
A coin is tossed 100 times and lands heads 46 times. Is the coin unfair?
- Experimental probability of heads = 46/100 = 0.46.
- Theory for a fair coin is 0.5, so the difference is 0.5 − 0.46 = 0.04.
- That is a small difference for 100 tosses, so there is no good reason to call the coin unfair.
Your go. The first step is given. A spinner is spun 40 times and lands on green 14 times. Write the experimental probability of green as a decimal and as a percentage.
- Green happened 14 times out of 40 spins: 14/40.
- Finish it, then check.
Check the answer
14/40 = 0.35, which is 35%.
Check before you go on
1. A coin is tossed 20 times and lands heads 7 times. What is the experimental probability of heads?
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 drawing pin is dropped 25 times and lands point up 10 times. Write the experimental probability of landing point up as a decimal.
3. A bus was late on 6 of the last 40 days. Write the experimental probability that it is late as a percentage. Type the number only.
Level 2 · Building
4. A spinner is spun 50 times: red 18 times, blue 12 times and green 20 times. Find the experimental probability of blue, as a decimal.
5. A fair coin is tossed 60 times and lands heads 33 times. Find the difference between the experimental probability of heads and the theoretical probability, as a decimal.
6. A fair die has shown no six in its last 8 rolls. What is the probability of a six on the next roll?
Level 3 · Stretch
7. A coin is tossed 20 times and lands heads 12 times. It is then tossed another 20 times. How many heads are needed in the second 20 tosses for the overall relative frequency of heads to be exactly 0.5?
8. In 80 trials an event happened with relative frequency 0.35. In 20 more trials it happened 10 times. What is the relative frequency over all 100 trials?
Something to think about
Two students each roll a die 20 times. Could one get 7 sixes and the other 1? Explain why that would not prove that one die is unfair. What could they do to find out?
One way to think about it
Yes, with only 20 rolls each, results far from 20/6 are possible for a fair die, though 7 is unusual: the figure's own run had 7 sixes in its first 20 rolls. Neither run proves anything. They could roll each die many more times and compare the relative frequencies: a fair die tends to settle near 1/6, a loaded one near something else.
Exit ticket
9. Group A tossed a coin 10 times and got 7 heads. Group B tossed the same coin 200 times and got 104 heads. Which result is the better estimate of the probability of heads, and why?