Chapter 8 of 8

All 8 chapters
  1. 1Collecting data, readunderstood
  2. 2Bar charts and pie charts, readunderstood
  3. 3Mean, median and mode, readunderstood
  4. 4Comparing two groups, readunderstood
  5. 5Line graphs and honest charts, readunderstood
  6. 6The language of chance, readunderstood
  7. 7Working out chances, readunderstood
  8. 8Experiments and fair games, readunderstood

M3.7.8 Chapter 8

Experiments and fair games

I tossed a coin 50 times. Why did I not get exactly 25 heads?

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

  • Toss a coin 50 times and compare with what you expected
  • Decide if a dice game is fair

Probability says what to expect. An experiment shows what really happens. A fair coin has a chance of 1/2 of landing on heads. So in 50 tosses we expect about 25 heads.

We say "about". Chance does not count tosses for us. A real run of 50 tosses can give 24, 26 or 28 heads. It gives exactly 25 only about 1 time in 9. Most runs land a few heads away from 25.

Draw it first

Toss a coin 50 times

  1. Step 1: a record of 50 coin tosses in 5 rows of 10. H is heads and T is tails. Step 2: the record is replaced by a tally of heads, 26, and a tally of tails, 24. Step 3: a label says expected 25, found 26.HTTHHTHHTHTHTTHHTHTTHHTHTHHTTTTHHTHTHHTHHTHTHTTHTH

    1 Toss a coin 50 times. Write H or T each time, in rows of ten.

  2. Step 1: a record of 50 coin tosses in 5 rows of 10. H is heads and T is tails. Step 2: the record is replaced by a tally of heads, 26, and a tally of tails, 24. Step 3: a label says expected 25, found 26.headstails

    2 Tally the heads and the tails. Count each tally.

  3. Step 1: a record of 50 coin tosses in 5 rows of 10. H is heads and T is tails. Step 2: the record is replaced by a tally of heads, 26, and a tally of tails, 24. Step 3: a label says expected 25, found 26.headstailsexpected 25, found 26

    3 Compare with what you expected. Expect 25 heads, because 50 × 1/2 = 25.

Worked example

A fair coin is tossed 50 times and gives 26 heads. How many heads would you expect? How far is the result from that?

  1. A fair coin has a chance of 1/2 of heads on each toss.
  2. Expect about 50 × 1/2 = 25 heads.
  3. The tosses gave 26 heads. That is 26 − 25 = 1 more than expected.

Answer: 25 heads expected. The result was 1 more.

Keep tossing and the share of heads usually settles closer to one half. Here is the share of heads after every ten tosses of the same 50.

A line graph of the share of heads. The axis labels are tosses so far and share of heads. The scale runs from 0 to 1 in steps of 0.1. The points after 10, 20, 30, 40 and 50 tosses are 0.6, 0.5, 0.5, 0.525, 0.52.00.10.20.30.40.50.60.70.80.911020304050tosses so farshare of heads
The share of heads starts at 0.6 and then stays close to 0.5.

Now think about games. A game is fair when every player has the same chance of winning. To check, count the outcomes that win for each player. Do not guess from how the rules sound.

Worked example

Ravi and Leila roll a dice. Ravi wins on 1 or 2. Leila wins on 3, 4, 5 or 6. Is the game fair? If not, how can you make it fair?

  1. A dice has 6 equally likely faces. Ravi wins on 1 and 2. That is 2 faces.
  2. Leila wins on 3, 4, 5 and 6. That is 4 faces.
  3. The chances are 2/6 and 4/6. They are not equal, so the game is not fair.
  4. To make it fair, give each player the same number of faces: 6 ÷ 2 = 3.

Answer: No. Ravi has 2 faces and Leila has 4. Each needs 3 faces.

The six faces of a dice, numbered 1 to 6. The faces 1 and 2 are hatched for Ravi. The faces 3, 4, 5 and 6 are plain, for Leila.123456
Ravi has 2 hatched faces. Leila has 4 plain faces. Leila is twice as likely to win.

Worked example

Amina and Kofi roll two dice and add the numbers. Amina wins if the sum is 7. Kofi wins if the sum is 9 or more. Is the game fair?

  1. Two dice have 6 × 6 = 36 equally likely outcomes. The grid shows the sum of each.
  2. Amina wins on a sum of 7. That is 6 outcomes.
  3. Kofi wins on a sum of 9, 10, 11 or 12: 4 + 3 + 2 + 1 = 10 outcomes.
  4. Amina has 6 chances in 36 and Kofi has 10. They are not equal, so the game is not fair.

Answer: No. Kofi has 10 chances in 36 and Amina has 6.

A grid of the sum of two dice. The labels are first and second dice. The numbers 1 to 6 run along the top and down the side. The 6 cells with a sum of 7 are hatched.second dicefirst123456123456234567345678456789567891067891011789101112
Amina's sums. Six cells out of 36 are hatched, so her chance is 6/36, which is 1/6.
A grid of the sum of two dice. The labels are first and second dice. The numbers 1 to 6 run along the top and down the side. The 10 cells with a sum of 9 or more are hatched.second dicefirst123456123456234567345678456789567891067891011789101112
Kofi's sums. Ten cells out of 36 are hatched, so his chance is 10/36, which is 5/18.

See it three ways

With things

Toss a coin 10 times and count the heads. You may get 6. You expected 5.

As a picture

A record of 10 coin tosses in one row. H is heads and T is tails. The tosses are H, T, T, H, H, T, H, H, T, H. There are 6 heads.HTTHHTHHTH

In symbols

Expected: 10 × 1/2 = 5 heads. Found: 6 heads. The difference is 6 − 5 = 1.

Try it

Your own 50 tosses

You need: a coin, paper and a pencil.

  1. Write "heads" and "tails" on your paper. Toss the coin 50 times.
  2. Make a tally mark for each result. Every fifth mark crosses the four before it.
  3. Count the heads and the tails. Add them. They must make 50.
  4. Compare the heads with 25. Is the difference small?

What you will see: A typical run gives about 25 heads and 25 tails, plus or minus a few. In the run shown, heads has 26 and tails has 24. Your numbers will differ. That is how chance works.

If you cannot try it, this is what happens.
A bar chart of 50 tosses. The axis label is tosses. The bars are heads 26 and tails 24. The scale runs from 0 to 30 in steps of 5.051015202530headstailstosses

A wrong idea many people have

If you toss a fair coin 50 times, you must get exactly 25 heads.

Why it feels right: A chance of one half sounds like a promise of half.

The test that breaks it: Toss a coin 10 times and count the heads. Do it again. You will not always get 5.

What is true: A probability says what is expected on average. Real results vary a little. With more tosses the share of heads usually gets closer to one half.

Key words

experiment
a test you really do and record, such as 50 tosses.
expect
what probability says should happen on average.
fair
a game where every player has the same chance of winning.

Back to your guess

The answer is A: Amina tossed a coin 50 times and got 28 heads. This is normal, because results of an experiment vary a little.

A fair coin lands on heads about half the time. That is what we expect. Chance does not follow a plan. Real results are usually close to 25 heads. Only about 1 run in 9 gives exactly 25.

Check yourself

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

  1. A coin is tossed 50 times and gives 28 heads. What does this show?

    Tap your answer.

    1. Not this one. A fair coin is not stuck on exactly half. 28 is close to the 25 we expect, so a fair coin often gives a result like this. Go to M3.7.8. Try another.

    2. Not this one. Nothing is wrong. Chance gives results near the expected number, not exactly on it. Go to M3.7.8. Try another.

    3. Yes, that is right. We expect about 25 heads. 28 is close to 25, so the coin is behaving like a fair coin.

    See the answer

    This is a normal result for a fair coin.

    We expect about 25 heads. 28 is close to 25, so the coin is behaving like a fair coin.

    • If you chose A: A fair coin is not stuck on exactly half. 28 is close to the 25 we expect, so a fair coin often gives a result like this. Go to M3.7.8.
    • If you chose B: Nothing is wrong. Chance gives results near the expected number, not exactly on it. Go to M3.7.8.

  2. A fair coin is tossed 100 times. About how many heads do you expect?

    Tap your answer.

    1. Not this one. That is a quarter of 100. A fair coin lands on heads half of the time, not a quarter. Go to M3.7.7. Try another.

    2. Yes, that is right. The chance of heads is 1/2. Half of 100 is 100 × 1/2 = 50.

    3. Not this one. That would be heads every time. Each toss has a chance of only 1/2 of heads. Go to M3.7.6. Try another.

    See the answer

    About 50 heads

    The chance of heads is 1/2. Half of 100 is 100 × 1/2 = 50.

    • If you chose A: That is a quarter of 100. A fair coin lands on heads half of the time, not a quarter. Go to M3.7.7.
    • If you chose C: That would be heads every time. Each toss has a chance of only 1/2 of heads. Go to M3.7.6.

  3. Two friends roll a dice. Which game is fair?

    Tap your answer.

    1. Yes, that is right. Even numbers are 2, 4 and 6. Odd numbers are 1, 3 and 5. Each player has 3 faces out of 6, so the chances are equal.

    2. Not this one. Ana wins on 2 faces and Ravi wins on 4 faces. Ravi has twice the chance, so it is not fair. Go to M3.7.8. Try another.

    3. Not this one. Ana has 1 face and Ravi has 5. Ravi is much more likely to win. Go to M3.7.8. Try another.

    See the answer

    Ana wins on an even number. Ravi wins on an odd number.

    Even numbers are 2, 4 and 6. Odd numbers are 1, 3 and 5. Each player has 3 faces out of 6, so the chances are equal.

    • If you chose B: Ana wins on 2 faces and Ravi wins on 4 faces. Ravi has twice the chance, so it is not fair. Go to M3.7.8.
    • If you chose C: Ana has 1 face and Ravi has 5. Ravi is much more likely to win. Go to M3.7.8.

  4. Mei rolls a dice. She wins on a 5 or a 6. The hatched faces are the faces she wins on. What is the chance that Mei wins?

    The six faces of a dice, numbered 1 to 6. The faces 5 and 6 are hatched. They are the faces Mei wins on.123456
    Tap your answer.

    1. Not this one. Half would be 3 faces out of 6. Mei wins on only 2 faces. Go to M3.7.7. Try another.

    2. Not this one. A dice has 6 faces, not 5. Divide the 2 winning faces by all 6 faces. Go to M3.7.7. Try another.

    3. Yes, that is right. Two faces out of 6 win. 2/6 in simplest form is 1/3.

    See the answer

    1/3

    Two faces out of 6 win. 2/6 in simplest form is 1/3.

    • If you chose A: Half would be 3 faces out of 6. Mei wins on only 2 faces. Go to M3.7.7.
    • If you chose B: A dice has 6 faces, not 5. Divide the 2 winning faces by all 6 faces. Go to M3.7.7.

  5. Sipho tosses a coin 10 times and gets 7 heads. Leila tosses a coin 100 times and gets 52 heads. Whose result is a better guide to the true chance of heads?

    Tap your answer.

    1. Not this one. A short run can wander far from the expected result. Ten tosses is too few to trust. Go to M3.7.8. Try another.

    2. Yes, that is right. With more tosses the share of heads usually gets closer to one half. Leila's share, 52 out of 100, is very close to it.

    3. Not this one. More tosses give a more reliable guide. 100 tosses tell you more than 10. Go to M3.7.8. Try another.

    See the answer

    Leila's, because more tosses give a better guide.

    With more tosses the share of heads usually gets closer to one half. Leila's share, 52 out of 100, is very close to it.

    • If you chose A: A short run can wander far from the expected result. Ten tosses is too few to trust. Go to M3.7.8.
    • If you chose C: More tosses give a more reliable guide. 100 tosses tell you more than 10. Go to M3.7.8.

  6. A fair coin is tossed 80 times. About how many heads do you expect?

    See the answer

    40 heads

    The chance of heads is 1/2. Half of 80 is 80 × 1/2 = 40.

  7. A coin is tossed 50 times and lands on heads 26 times. What share of the tosses were heads? Write it as a decimal.

    See the answer

    0.52

    Divide the heads by the tosses: 26 ÷ 50 = 0.52.

  8. From an earlier chapter

    Review: a bag holds 7 beads of the same size and 2 are red. What is the chance of not picking red?

    See the answer

    5/7

    The beads that are not red are 7 − 2 = 5. The chance is 5/7.

Keep it

Copy this into your exercise book.

Expect about half heads from a fair coin, but results vary a little. More tosses usually come closer. A game is fair when each player has the same chance. Count the winning outcomes.

Draw: Draw the grid of two dice sums with the sums of 7 hatched. Write 6 out of 36.

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 (968 KB). It opens on any phone.