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Experimental and theoretical probability

What the math predicts against what actually happened, and why they differ.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

In this lesson you compare the probability worked out from the setup with the fraction of times an event actually happened. They rarely match exactly, and they get closer the more trials you do, which is the law of large numbers. You will also use either kind of probability to predict how often something will happen.

2. What you already know

In the last lesson you learned that a probability is a number from 0 to 1, and that when outcomes are equally likely you can find it by counting: favorable outcomes over all outcomes. You can also multiply a fraction or a decimal by a whole number, such as $0.25 \times 80 = 20$. In this lesson you find probabilities a second way, by doing an experiment and counting what actually happens, and you compare the two.

3. Words in this lesson

4. Two ways to find a probability

There are two ways to find a probability.

• Theoretical probability comes from thinking about the setup. A fair die has 6 equally likely faces, so the probability of rolling a 4 is $\frac{1}{6}$. You do not roll anything.
• Experimental probability comes from doing the experiment and counting:

$$P(\text{event}) \approx \frac{\text{number of times it happened}}{\text{number of trials}}$$

If you roll the die 60 times and see a 4 on 13 of them, the experimental probability is $\frac{13}{60} \approx 0.22$.

The two rarely match exactly. Chance makes results wobble, so 60 rolls will seldom give exactly ten 4s. But as the number of trials grows, the experimental probability tends to settle closer and closer to the theoretical one. This is the law of large numbers.

Either kind of probability can be used to predict: multiply the probability by the number of trials to get the expected count.

5. Why results wobble: the law of large numbers

Suppose a class flips a fair coin and keeps a running count. Here is one real set of results a class might get.

Notice two things. First, the number of heads is almost never exactly half. After 10,000 flips it is 12 short of 5,000. Second, the fraction of heads gets closer and closer to 0.5, because 12 out of 10,000 is a tiny share.

So a short experiment can be far from the theory without anything being wrong. Seven heads in ten flips is not surprising. Seven thousand heads in ten thousand flips would be very surprising, and would make you suspect the coin.

6. Making predictions: probability times trials

To predict how often something will happen, multiply its probability by the number of trials.

• A fair die is rolled 300 times. The probability of a 6 is $\frac{1}{6}$, so you expect about $\frac{1}{6} \times 300 = 50$ sixes.
• A spinner with 5 equal sections is spun 80 times. You expect each section about $\frac{1}{5} \times 80 = 16$ times.
• A soccer goalkeeper has stopped 30% of penalty kicks in the past. In the next 20 kicks you expect about $0.3 \times 20 = 6$ saves.

The word about matters. The expected count is a prediction of the typical result, not a promise. If you roll the die 300 times, 46 or 55 sixes would be perfectly normal.

7. When there is no theory: building a model from data

Some experiments have no equally likely outcomes to count. A thumbtack can land point up or on its side, but its shape makes one outcome more likely, and nobody can work out by how much just by looking. A paper cup can land on its top, its bottom or its side.

For these, the experiment is the model. Drop the cup 200 times, count each outcome and divide. If it lands on its side 130 times, on its bottom 46 times and on its top 24 times, the model is: side 0.65, bottom 0.23, top 0.12. The three probabilities add to 1, as every model's must.

A model like this is called not uniform, because the outcomes have different probabilities. A fair die is a uniform model: every outcome has the same probability. Both kinds let you make predictions.

8. Theory for unequal sections: use the angle

Some spinners and prize wheels have sections of different sizes. You can still find a theoretical probability, because a fair wheel is equally likely to stop at any point around its edge. The probability of a section is the fraction of the full turn it covers.

A full turn is 360°. A section with an angle of 90° at the center covers $\frac{90}{360} = \frac{1}{4}$ of the wheel, so its probability is 0.25. A section of 30° has probability $\frac{30}{360} = \frac{1}{12}$. If a wheel has one 180° section, one 120° section and one 60° section, the model is $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$, and these add to 1.

Predictions work exactly as before. Spin that wheel 120 times and you expect the 60° section about $\frac{1}{6} \times 120 = 20$ times. Then you can spin it, count, and compare, just as you would with equal sections.

9. Is the model wrong? Deciding when a gap is too big

If an experiment disagrees with the theory, there are two possible reasons. Either chance produced an unusual run, or the theory does not fit, perhaps because the die is weighted or the spinner sticks.

Ask two questions. How many trials were there? And how big is the gap compared with the gaps in the other outcomes? A gap of 3 in 20 trials means little. A gap of 30 in 200 trials, when every other outcome is within 5 of its prediction, is a strong hint that something is unfair. The honest next step is to do more trials: if the gap is real, it will still be there.

10. The method, step by step, and how to check it

Every problem in this lesson uses some of these steps.

1. Model: work out the theoretical probability if the outcomes are equally likely, or use the angle or area if they are not.
2. Predict: multiply the probability by the number of trials.
3. Count: record how many times the event actually happened.
4. Divide: times it happened over number of trials gives the experimental probability.
5. Compare: find the gap between experiment and theory.
6. Judge: a small gap, or a short experiment, is chance. A large gap that lasts over many trials means the model should change.

Each step has a reason. The prediction comes before the experiment so that the result cannot change what you expected. Dividing by the trials turns a count into a fraction, which can be compared across experiments of any size.

How to check. Every probability, theoretical or experimental, must be between 0 and 1. The probabilities in a model must add to 1. An expected count cannot be more than the number of trials. And if you predicted 50 and counted 48, do not call the theory wrong: that is exactly how chance behaves.

11. In the world: famous coin-flipping experiments

People have tested the law of large numbers by hand. In the 1700s the French scientist Buffon flipped a coin 4,040 times and got 2,048 heads, an experimental probability of $2{,}048 \div 4{,}040 \approx 0.507$. Around 1900 the English statistician Karl Pearson flipped 24,000 times and got 12,012 heads, which is $12{,}012 \div 24{,}000 = 0.5005$. During World War II, John Kerrich flipped a coin 10,000 times while held in a prison camp in Denmark and got 5,067 heads, 0.5067. None of them got exactly half. Pearson was 12 heads over, yet his experimental probability was within 0.0005 of the theory, closer than Buffon's shorter run.

12. In the world: a batting average is an experimental probability

A baseball batting average is hits divided by at bats. A player with 150 hits in 500 at bats has an average of $150 \div 500 = 0.300$, which fans say as three hundred. There is no theory that says what a batter's chance of a hit should be, so the season's record is the model. A manager can use it to predict: in the next 40 at bats, this player should get about $0.3 \times 40 = 12$ hits. Early in the season, averages jump around because there are few at bats. A player with 4 hits in his first 8 at bats is batting 0.500, but nobody expects that to last over 500 at bats.

13. In the world: testing seeds before planting

Farmers and seed companies test seeds before selling or planting them. Suppose a company plants 200 seeds from a large batch of tomato seeds, and 176 of them sprout. The experimental probability that a seed sprouts is $176 \div 200 = 0.88$, so the company prints 88% germination on the packet. A gardener who plants 50 of those seeds can expect about $0.88 \times 50 = 44$ seedlings. If she needs 40 plants, she knows 50 seeds should be enough, with a few to spare in case her batch does a little worse than the test.

14. Mistakes to avoid

The most common mistake is expecting the experiment to match the theory exactly. Twelve heads in twenty flips is not proof that the coin is unfair. Short experiments wobble.

The second is the opposite: thinking a coin owes you tails after a run of heads. A fair coin has no memory. After five heads in a row, the next flip is still 0.5 heads. The law of large numbers works by the many later flips swamping the early ones, not by the coin correcting itself.

The third is trusting an experiment with very few trials. An estimate from 10 trials can easily be off by 0.2; one from 1,000 trials rarely is.

The last is dividing by the wrong number: the bottom of an experimental probability is the number of trials, not the number of times the event did not happen.

15. How a paper cup lands

1. A cup is tossed 50 times. It lands on its side 34 times, open end up 10 times and open end down 6 times. Check the count of trials.

   $34 + 10 + 6 = 50$

   Every toss must be counted exactly once.

2. Write the experimental probability of landing on its side.

   $P(\text{side}) = \tfrac{34}{50}$

   Times it happened over the number of trials.

3. Change it to a decimal.

   $\tfrac{34}{50} = \tfrac{68}{100} = 0.68$

   Doubling the top and bottom gives hundredths.

4. Find the other two experimental probabilities.

   $P(\text{up}) = \tfrac{10}{50} = 0.2, \quad P(\text{down}) = \tfrac{6}{50} = 0.12$

   Each outcome gets its own fraction of the 50 trials.

5. Check that the model adds to 1.

   $0.68 + 0.2 + 0.12 = 1$

   The three outcomes cover every toss, so their probabilities must add to 1.

16. A die rolled 60 times

1. Find the theoretical probability of rolling a 3 on a fair die.

   $P(3) = \tfrac{1}{6}$

   One of six equally likely faces shows a 3.

2. Predict the number of 3s in 60 rolls.

   $\tfrac{1}{6} \times 60 = 10$

   Expected count is probability times the number of trials.

3. The die is rolled 60 times and shows a 3 on 13 rolls. Write the experimental probability.

   $P(3) \approx \tfrac{13}{60}$

   Times it happened over the number of trials.

4. Change both probabilities to decimals.

   $\tfrac{13}{60} \approx 0.22, \qquad \tfrac{1}{6} \approx 0.17$

   Decimals are easy to compare.

5. Find the gap in counts.

   $13 - 10 = 3$

   The experiment gave 3 more 3s than predicted.

6. Judge the gap.

   $3 \text{ out of } 60 \Rightarrow \text{chance}$

   A gap of a few in only 60 rolls is normal, so there is no reason to think the die is unfair.

17. Is this spinner fair?

1. A spinner has four equal sections: red, blue, green and yellow. Write the theoretical probability of each color.

   $P = \tfrac{1}{4} = 0.25$

   Four equal sections are equally likely.

2. Predict each color's count in 200 spins.

   $0.25 \times 200 = 50$

   Expected count is probability times the number of trials.

3. In 200 spins red came up 74 times. Find red's experimental probability.

   $\tfrac{74}{200} = 0.37$

   Times it happened over the number of trials.

4. Blue came up 42 times. Find its experimental probability.

   $\tfrac{42}{200} = 0.21$

   Each color is compared with its own prediction.

5. Green came up 44 times and yellow 40 times. Find theirs.

   $\tfrac{44}{200} = 0.22, \qquad \tfrac{40}{200} = 0.2$

   The same division for each color.

6. Check that the counts cover every spin.

   $74 + 42 + 44 + 40 = 200$

   Every spin landed on exactly one color.

7. Compare each count with the prediction of 50.

   $+24, \quad -8, \quad -6, \quad -10$

   Red is 24 above its prediction, much further off than any other color.

8. Draw a conclusion, and plan the next step.

   $0.37 \gg 0.25 \Rightarrow \text{probably not fair}$

   A gap that big in 200 spins is unlikely to be chance, so spin it many more times to be sure.

18. Your turn: a coin flipped 400 times gives 212 heads

1. Predict the number of heads.

   $0.5 \times 400 = 200$

   Expected count is probability times the number of trials.

2. Your turn: work this step out. Its working is at the end of the packet.

   Find the experimental probability of heads.

3. Your turn: work this step out. Its working is at the end of the packet.

   Find the gap between experiment and theory.

19. Guided practice

A coin was flipped $80$ times and came up heads $49$ times, an experimental probability of $0.6125$. The theoretical probability of heads is 0.5. What is the best explanation?

• Results vary by chance; with many more flips the experimental probability would move closer to 0.5
• The coin must be unfair, because $49$ is not half of $80$
• The theoretical probability is wrong and should be $0.6125$

20. Guided practice

A spinner with four equal sections, one of them red, is spun $80$ times and lands on red $23$ times. Complete the worked solution comparing the experiment with the theory.

1. Write the theoretical probability of red.

   $P(\text{red}) = \tfrac{1}{4} = 0.25$

   Red is one of four equally likely sections.

2. Multiply by the number of spins to find the expected count.

   $0.25 \times 80 =$ _e

   Expected count is probability times the number of trials.

3. Divide the observed count by the number of spins.

   $\text{experimental } P(\text{red}) =$ _p

   Experimental probability is the fraction of trials where red happened.

4. Subtract the expected count from the observed count.

   $\text{observed} - \text{expected} =$ _g

   A small gap is what chance usually produces.

21. Guided practice

Ana and Ben each drop a paper cup to estimate the probability that it lands on its side. Ana drops it $10$ times and Ben drops it $300$ times. Whose experimental probability is likely to be closer to the true probability?

• Ana's
• Ben's
• Both are equally reliable

22. Practice

A thumbtack was dropped $80$ times and landed point up $4$ times. What is the experimental probability that it lands point up? Give a fraction or a decimal.

$P(\text{point up}) =$ _answer

23. Practice

A spinner has $7$ equal sections. It is spun $154$ times. About how many times would you expect it to land on one particular section?

about _answer times

24. Practice

A basketball player has made $45$ of her $50$ free throws this season. What is her experimental probability of making a free throw, as a decimal, and about how many of her next $20$ free throws would you predict she makes?

Experimental probability: _r. Predicted makes: _m

25. Somewhere new

A carnival game has a bag of $60$ colored tiles, but you cannot see inside. You draw a tile, note its color and put it back, $30$ times in all. Red comes up $9$ times. About how many of the tiles in the bag are red?

Answer:

26. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

27. Test question

A prize wheel at a fair has sections of different sizes. The blue section's angle at the center is $144^\circ$. Over one day the wheel is spun $360$ times. What is the probability of blue, and about how many times would you expect it to stop on blue?

$P(\text{blue}) =$ _p. Expected stops on blue: _e

28. What you can do now

You can compare experimental and theoretical probability. Without looking: if a fair coin gives 12 heads in 20 flips, is it unfair, and what would make you more sure? How many 6s do you expect in 120 rolls of a die?

Working for the steps left to you

18. Your turn: a coin flipped 400 times gives 212 heads, step 2

18. Your turn: a coin flipped 400 times gives 212 heads, step 3