Chapter 2 of 8

All 8 chapters
  1. 1Listing all the outcomes, readunderstood
  2. 2Relative frequency and expected results, readunderstood
  3. 3The OR rule and the AND rule, readunderstood
  4. 4Tree diagrams, readunderstood
  5. 5Picking without putting back, readunderstood
  6. 6Venn diagrams and sets, readunderstood
  7. 7Conditional probability, readunderstood
  8. 8Counting choices, readunderstood

M4.6.2 Chapter 2

Relative frequency and expected results

A bottle top is dropped on a table. How can I find the chance that it lands top up?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Estimate a probability from the results of an experiment
  • Find how many times to expect an outcome in 200 tries

Some events have no list of equally likely outcomes. A bottle top lands top up or the other way up, but the two are not equally likely. The only way to learn the chance is to try it.

Each go is a trial. The relative frequency of an event is the number of times it happened divided by the number of trials. It is an estimate of the probability.

Draw it first

From a tally to an estimate

  1. A table with the head drops, top up, estimate. Step 1 has one row, 50 drops and 18 top up, with the last cell empty. Step 2 writes 18 ÷ 50 in the last cell. Step 3 writes 0.36 there.dropstop upestimate5018

    1 Drop the bottle top 50 times. Count the drops that land top up. Here it is 18.

  2. A table with the head drops, top up, estimate. Step 1 has one row, 50 drops and 18 top up, with the last cell empty. Step 2 writes 18 ÷ 50 in the last cell. Step 3 writes 0.36 there.dropstop upestimate501818 ÷ 50

    2 Divide the number of times it happened by the number of trials: 18 ÷ 50.

  3. A table with the head drops, top up, estimate. Step 1 has one row, 50 drops and 18 top up, with the last cell empty. Step 2 writes 18 ÷ 50 in the last cell. Step 3 writes 0.36 there.dropstop upestimate50180.36

    3 The result, 0.36, is the relative frequency. It is your estimate of the probability.

Worked example

A bottle top is dropped 200 times and lands top up 73 times. Estimate the probability that it lands top up.

  1. Relative frequency = number of times it happened ÷ number of trials.
  2. It happened 73 times in 200 trials.
  3. 73 ÷ 200 = 0.365.
  4. The estimate of the probability is 0.365.

Answer: 0.365

One experiment gives one estimate. Here are the same bottle top's results after more and more drops. The estimates move around, then settle near 0.37.

A table with the head drops, top up, estimate. Rows: 10, 4, 0.4; 20, 9, 0.45; 50, 18, 0.36; 100, 37, 0.37; 200, 73, 0.365; 500, 184, 0.368.dropstop upestimate1040.42090.4550180.36100370.37200730.3655001840.368
The estimate is the last column. Early estimates swing. Later ones settle.

A fair coin shows the same thing, and here we know the true probability is 0.5. This record holds the first 40 of 100 tosses. H is a head and T is a tail.

A record of the first 40 tosses of a coin in four rows of ten. A head is a white circle marked H. A tail is a hatched circle marked T. The tosses run left to right, row by row, in the order they happened.HHHTHHTTHHTTTTHTTTHTHHHTHTTTTHHHHTTTHTTH
The first 40 tosses, in order. Heads came up 19 times.

Now work out the relative frequency of heads after 10 tosses, 20 tosses, and so on to 100. Plot each one against the number of tosses.

A graph with the x axis tosses from 0 to 100 and the y axis relative frequency from 0 to 1. Ten dots joined by lines show the relative frequency of heads after 10, 20 up to 100 tosses. The first dot is high at 0.7, the next dips below 0.5, and the later dots stay close to a dashed line named 0.5.0.5204060801000.250.50.7510tossesrelative frequency
With few tosses the estimate swings. With more tosses it stays close to 0.5.

More trials usually give a better estimate. But a relative frequency never proves the probability. After 100 tosses the coin shows 0.52, not 0.5, and a new set of 100 tosses would give a different number.

Now turn the idea round. If you know the probability, you can say what to expect. The expected number of times an event happens in n trials is the probability times n.

Worked example

A spinner lands on blue with probability 3/10. It is spun 200 times. How many times should you expect blue?

  1. Expected number = probability × number of trials.
  2. Here that is 3/10 × 200.
  3. 200 ÷ 10 = 20.
  4. 3 × 20 = 60.
  5. You expect blue about 60 times.

Answer: 60

A bar chart called heads. Two bars. The bar named expected is 100 high, which is 200 tosses times 0.5. The bar named actual is 93 high.020406080100120expectedactualheads
A fair coin tossed 200 times: 100 heads expected, 93 found. That is normal.

Worked example

A fair dice is rolled 200 times. How many sixes should you expect?

  1. The probability of a six is 1/6.
  2. Expected number = 1/6 × 200 = 200 ÷ 6. To 3 decimal places, 200 ÷ 6 = 33.333.
  3. To 1 decimal place, 200 ÷ 6 = 33.3.
  4. To the nearest whole number, you expect 33 sixes.
  5. The expected number need not be a whole number, because it is an average over many sets of 200 rolls.

Answer: 33.3

Try it

Toss a coin and watch the estimate settle

You need: a coin, or a flat stone with a mark on one side, a pencil and paper.

  1. Toss the coin 100 times. Write H or T each time.
  2. After every 10 tosses, work out the relative frequency of heads so far.
  3. Plot each value on a graph with tosses across and relative frequency up.
  4. Compare your points at 10, 50 and 100 tosses. Which are closest to 0.5?

What you will see: The early points jump about. The later points move less and stay near 0.5. Nobody gets exactly 0.5 every time.

If you cannot try it, this is what happens.
A table with the head tosses, heads, estimate. Rows: 10, 7, 0.7; 50, 26, 0.52; 100, 52, 0.52.tossesheadsestimate1070.750260.52100520.52

A wrong idea many people have

The expected number is how many will happen.

Why it feels right: The word "expect" sounds like a promise, and 200 × 1/2 gives one clear number, 100.

The test that breaks it: Toss a fair coin 200 times, then do it again. The heads will be near 100 each time, but not exactly 100 each time.

What is true: The expected number is an average. The real number is usually close to it and often a little different.

Key words

trial
one go of an experiment, such as one toss of a coin.
relative frequency
the number of times an event happened divided by the number of trials. It estimates the probability.
expected number
the probability times the number of trials. It is an average, not a promise.

Back to your guess

The answer is B: Drop it many times and see what fraction of the drops land top up.

The two ways are not equally likely, because the top and the bottom are different shapes. Counting outcomes does not work. Instead, drop it many times. The fraction of drops that land top up is an estimate of the probability.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A bottle top lands top up 30 times in 100 drops. What is the relative frequency of landing top up?

    Tap your answer.

    1. Not this one. That is the relative frequency of landing the other way up, 70 out of 100. Go to M4.6.2. Try another.

    2. Not this one. That is the number of times, not a fraction of the drops. Divide it by the 100 drops. Go to M4.6.2. Try another.

    3. Yes, that is right. Relative frequency = 30 ÷ 100 = 0.3.

    See the answer

    0.3

    Relative frequency = 30 ÷ 100 = 0.3.

    • If you chose A: That is the relative frequency of landing the other way up, 70 out of 100. Go to M4.6.2.
    • If you chose B: That is the number of times, not a fraction of the drops. Divide it by the 100 drops. Go to M4.6.2.

  2. You want a good estimate of the chance that a bottle top lands top up. Which plan is best?

    Tap your answer.

    1. Yes, that is right. More trials usually give an estimate nearer the true probability. The estimates for 10 or 20 drops swing more than the estimate for 500.

    2. Not this one. A feeling is not an experiment, and a few drops are too few to settle near the true value. Go to M4.6.2. Try another.

    3. Not this one. Stopping when the number looks right chooses the answer you wanted. Decide the number of trials first. Go to M4.6.2. Try another.

    See the answer

    Drop it as many times as you can, and divide.

    More trials usually give an estimate nearer the true probability. The estimates for 10 or 20 drops swing more than the estimate for 500.

    • If you chose B: A feeling is not an experiment, and a few drops are too few to settle near the true value. Go to M4.6.2.
    • If you chose C: Stopping when the number looks right chooses the answer you wanted. Decide the number of trials first. Go to M4.6.2.

  3. A spinner lands on green with probability 1/4. It is spun 200 times. How many times should you expect green?

    Tap your answer.

    1. Not this one. You used the bottom of the fraction. Expected number = probability × trials = 1/4 × 200. Go to M4.6.2. Try another.

    2. Not this one. That is the number to expect for not green, 3/4 of 200. Go to M4.6.2. Try another.

    3. Not this one. You divided 200 by 1/4. Multiply the probability by the number of trials instead. Go to M4.6.2. Try another.

    4. Yes, that is right. Expected number = 1/4 × 200 = 50.

    See the answer

    50

    Expected number = 1/4 × 200 = 50.

    • If you chose A: You used the bottom of the fraction. Expected number = probability × trials = 1/4 × 200. Go to M4.6.2.
    • If you chose B: That is the number to expect for not green, 3/4 of 200. Go to M4.6.2.
    • If you chose C: You divided 200 by 1/4. Multiply the probability by the number of trials instead. Go to M4.6.2.

  4. A coin is tossed 200 times and lands heads 106 times. What does this tell you?

    Tap your answer.

    1. Not this one. A fair coin rarely gives exactly half. The difference here is small, and chance explains it. Go to M4.6.2. Try another.

    2. Yes, that is right. For a fair coin the expected number is 1/2 × 200 = 100. It is an average over many sets of tosses, so real results are often a little off. 106 is a small difference.

    3. Not this one. Coins do not make up for earlier tosses. Each toss of a fair coin is a new chance of 1/2. Go to M4.6.2. Try another.

    See the answer

    It is close to the 100 heads that a fair coin is expected to give, so a small difference is normal.

    For a fair coin the expected number is 1/2 × 200 = 100. It is an average over many sets of tosses, so real results are often a little off. 106 is a small difference.

    • If you chose A: A fair coin rarely gives exactly half. The difference here is small, and chance explains it. Go to M4.6.2.
    • If you chose C: Coins do not make up for earlier tosses. Each toss of a fair coin is a new chance of 1/2. Go to M4.6.2.

  5. Which statement about relative frequency is true?

    Tap your answer.

    1. Not this one. It is only an estimate, and 100 trials still leave room for chance. The coin gave 0.52 after 100 tosses. Go to M4.6.2. Try another.

    2. Not this one. It is near 0.5 for a fair coin with many tosses. It is rarely exactly 0.5. Go to M4.6.2. Try another.

    3. Yes, that is right. Relative frequency estimates the probability. More trials usually give a better estimate, but never a proof.

    See the answer

    It is an estimate that usually gets closer to the probability as the trials increase.

    Relative frequency estimates the probability. More trials usually give a better estimate, but never a proof.

    • If you chose A: It is only an estimate, and 100 trials still leave room for chance. The coin gave 0.52 after 100 tosses. Go to M4.6.2.
    • If you chose B: It is near 0.5 for a fair coin with many tosses. It is rarely exactly 0.5. Go to M4.6.2.

  6. From an earlier chapter

    Two dice are rolled and added. What is the probability that the total is 4?

    See the answer

    1/12

    A total of 4 is 1 and 3, 2 and 2, 3 and 1. That is 3 of 36 equally likely pairs, and 3/36 = 1/12.

  7. A bottle top lands top up 45 times in 250 drops. Estimate the probability that it lands top up.

    See the answer

    0.18

    Relative frequency = 45 ÷ 250 = 0.18.

  8. A spinner lands on red with probability 3/5. It is spun 150 times. How many times should you expect red?

    See the answer

    90

    Expected number = 3/5 × 150 = 90.

Keep it

Copy this into your exercise book.

Relative frequency = times it happened ÷ trials. It estimates a probability, and more trials usually improve it. Expected number = probability × trials, an average and not a promise.

Draw: A graph of relative frequency against trials, swinging at first and then settling near a dashed line.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (868 KB). It opens on any phone.