Chapter 1 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

M4.6.1 Chapter 1

Listing all the outcomes

I roll two dice and add the scores. Which total comes up most often?

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

  • List all the outcomes of two events in a table
  • Find a probability from a table of outcomes

An outcome is one possible result, such as a 4 on a dice. An event is the result you are interested in, such as "an even score". It can hold more than one outcome.

When every outcome is equally likely, the probability of an event is a count divided by a count. It is the number of outcomes in the event, divided by the number of outcomes in all. For an even score on a dice, that is 3/6, which is 1/2.

Six dice faces, 1 to 6, each with its number written under it. The faces 2, 4 and 6 are hatched, and the caption says even scores.123456even scores
Three of the six equally likely scores are even.

Now two things happen together: a coin is tossed and a dice is rolled. To count the outcomes without missing any, list them in a table. Put the coin outcomes down the side and the dice scores across the top.

Draw it first

List every outcome of a coin and a dice

  1. Step 1 lists the outcomes of each thing: a coin with two circles marked H and T, labelled coin, and a dice with its six faces, labelled dice. Step 2 is a table with H and T down the side and 1 to 6 across the top. Each cell holds its pair such as H1, and the cells H2, H4 and H6 are hatched.coinHTdice123456

    1 List the outcomes of each thing. A coin has two: H for head and T for tail. A dice has six: the scores 1 to 6.

  2. Step 1 lists the outcomes of each thing: a coin with two circles marked H and T, labelled coin, and a dice with its six faces, labelled dice. Step 2 is a table with H and T down the side and 1 to 6 across the top. Each cell holds its pair such as H1, and the cells H2, H4 and H6 are hatched.dicecoin123456HTH1H2H3H4H5H6T1T2T3T4T5T6

    2 Make a table. Every coin outcome meets every score in one cell. Hatch the cells that fit the event: a head and an even score.

A table with the coin down the side, H and T, and the dice across the top, 1 to 6. Each cell holds its pair such as H1. The cells H2, H4 and H6 are hatched.dicecoin123456HTH1H2H3H4H5H6T1T2T3T4T5T6
12 outcomes. The 3 hatched cells are a head and an even score.

Worked example

A coin is tossed and a dice is rolled. What is the probability of a head and an even score?

  1. The coin has 2 outcomes and the dice has 6. Each coin outcome goes with each score.
  2. So there are 2 × 6 = 12 outcomes in all, and they are equally likely.
  3. The cells with a head and an even score are H2, H4 and H6. That is 3 outcomes.
  4. P(head and even) = 3/12 = 1/4.

Answer: 1/4

The same table works for two dice. The first dice goes across the top and the second dice goes down the side. Each cell shows the total of its two scores. There are 36 cells.

A grid of two dice. The scores of dice 1, 1 to 6, run across the top, and the scores of dice 2, 1 to 6, run down the side. Each of the 36 cells shows the total of its two scores. The 6 cells with a total of 7 are hatched.dice 1dice 2123456123456234567345678456789567891067891011789101112
A total of 7 appears in 6 of the 36 cells.

The 36 pairs are equally likely, because each dice is fair and one dice cannot change the other. But the 11 totals from 2 to 12 are not equally likely. A total of 7 fills 6 cells. A total of 2 fills only one cell.

A bar chart called ways. The bars are the totals 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, and the height of each is the number of the 36 cells with that total. The bars are 1, 2, 3, 4, 5, 6, 5, 4, 3, 2 and 1 high, so the tallest bar is the total 7.012345623456789101112ways
The number of ways to make each total. The most likely total is 7.

Worked example

Two dice are rolled and added. What is the probability that the total is 10 or more?

  1. Use the table of 36 equally likely pairs.
  2. A total of 10 appears 3 times, 11 appears 2 times and 12 appears once.
  3. That is 3 + 2 + 1 = 6 cells.
  4. P(10 or more) = 6/36 = 1/6.

Answer: 1/6

A table with the head total, ways. Rows: 10, 3; 11, 2; 12, 1; 10 or more, 6.totalways10311212110 or more6
Three totals make 10 or more, and together they fill 6 cells.

Try it

Roll two dice 36 times

You need: two dice, or two sets of six pieces of paper numbered 1 to 6, each set in its own bag, a pencil and paper for a tally.

  1. Roll both dice and add the scores. Make a mark in the tally against that total.
  2. Do this 36 times.
  3. Draw a bar for each total.
  4. Compare with the bar chart of ways. Do not expect them to match exactly.

What you will see: The middle totals, 6, 7 and 8, come up most often. The totals 2 and 12 come up least. Your bars will not match the ways exactly, because the rolls are chance.

If you cannot try it, this is what happens.
A bar chart called rolls. The bars are the totals 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Their heights are 1, 2, 3, 4, 5, 7, 5, 3, 3, 2 and 1 high, from one learner who rolled two dice 36 times. The tallest bar is the total 7.0246823456789101112rolls

A wrong idea many people have

There are 11 totals, so each total has a probability of 1/11.

Why it feels right: The totals 2 to 12 are 11 results, and it seems fair to give each one the same share.

The test that breaks it: List the pairs. A total of 12 needs a 6 and a 6, so it is 1 of 36. A total of 7 can be made six ways, so it is 6 of 36.

What is true: Count only outcomes that are equally likely. The 36 pairs are. The 11 totals are not.

Key words

outcome
one possible result, such as a 5 on a dice or a tail on a coin.
event
the result you are interested in. It is made of one or more outcomes.
equally likely
having the same chance of happening, like the six faces of a fair dice.
probability
a number from 0 to 1 that measures how likely an event is.

Back to your guess

The answer is B: A total of 7, because more pairs of scores make it.

There are 36 pairs of scores, and all 36 are equally likely. A total of 7 can be made 6 ways. A total of 2 can be made only 1 way, with a 1 and a 1. So 7 comes up most often.

Check yourself

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

  1. A coin is tossed and a spinner with 4 equal sections, numbered 1 to 4, is spun. How many outcomes are there in all?

    Tap your answer.

    1. Not this one. You added 2 and 4. Each coin result goes with each spinner result, so multiply: 2 × 4. Go to M4.6.1. Try another.

    2. Yes, that is right. The coin has 2 outcomes and the spinner has 4. Each coin outcome goes with each spinner outcome, so there are 2 × 4 = 8.

    3. Not this one. You counted only the spinner. The coin has 2 outcomes too, so every spinner result comes twice. Go to M4.6.1. Try another.

    See the answer

    8

    The coin has 2 outcomes and the spinner has 4. Each coin outcome goes with each spinner outcome, so there are 2 × 4 = 8.

    • If you chose A: You added 2 and 4. Each coin result goes with each spinner result, so multiply: 2 × 4. Go to M4.6.1.
    • If you chose C: You counted only the spinner. The coin has 2 outcomes too, so every spinner result comes twice. Go to M4.6.1.

  2. A coin is tossed and a dice is rolled. What is the probability of a tail and a 5?

    Tap your answer.

    1. Not this one. You used only the dice. The coin must also show a tail, which is half of the cases. Go to M4.6.1. Try another.

    2. Not this one. You used only the coin. The dice must also show a 5, and that is 1 case in 6. Go to M4.6.1. Try another.

    3. Yes, that is right. There are 12 equally likely outcomes. Only T5 is a tail and a 5, so the probability is 1 out of 12.

    See the answer

    1/12

    There are 12 equally likely outcomes. Only T5 is a tail and a 5, so the probability is 1 out of 12.

    • If you chose A: You used only the dice. The coin must also show a tail, which is half of the cases. Go to M4.6.1.
    • If you chose B: You used only the coin. The dice must also show a 5, and that is 1 case in 6. Go to M4.6.1.

  3. Two dice are rolled and added. Use the table. What is the probability that the total is 8?

    A grid of two dice. The scores of dice 1, 1 to 6, run across the top, and the scores of dice 2, 1 to 6, run down the side. Each of the 36 cells shows the total of its two scores. The 5 cells with a total of 8 are hatched.dice 1dice 2123456123456234567345678456789567891067891011789101112
    Tap your answer.

    1. Yes, that is right. A total of 8 is 2 and 6, 3 and 5, 4 and 4, 5 and 3, 6 and 2. That is 5 cells of 36.

    2. Not this one. You counted the 11 totals as if they were equally likely. They are not. Count the cells of the 36 pairs. Go to M4.6.1. Try another.

    3. Not this one. You used the total, 8, as the number of favourable outcomes. Count the hatched cells instead. Go to M4.6.1. Try another.

    See the answer

    5/36

    A total of 8 is 2 and 6, 3 and 5, 4 and 4, 5 and 3, 6 and 2. That is 5 cells of 36.

    • If you chose B: You counted the 11 totals as if they were equally likely. They are not. Count the cells of the 36 pairs. Go to M4.6.1.
    • If you chose C: You used the total, 8, as the number of favourable outcomes. Count the hatched cells instead. Go to M4.6.1.

  4. Two coins are tossed. Which list shows every outcome?

    Tap your answer.

    1. Not this one. These are only two of the outcomes. A head with a tail can happen in two ways. Go to M4.6.1. Try another.

    2. Not this one. HT and TH are different outcomes. In HT the first coin is a head, and in TH the second coin is a head. Go to M4.6.1. Try another.

    3. Not this one. This lists one coin. Two coins make pairs, one result from each. Go to M4.6.1. Try another.

    4. Yes, that is right. The first coin can be H or T, and with each the second coin can be H or T. That gives 4 outcomes.

    See the answer

    HH, HT, TH, TT

    The first coin can be H or T, and with each the second coin can be H or T. That gives 4 outcomes.

    • If you chose A: These are only two of the outcomes. A head with a tail can happen in two ways. Go to M4.6.1.
    • If you chose B: HT and TH are different outcomes. In HT the first coin is a head, and in TH the second coin is a head. Go to M4.6.1.
    • If you chose C: This lists one coin. Two coins make pairs, one result from each. Go to M4.6.1.

  5. Two coins are tossed. What is the probability of one head and one tail?

    Tap your answer.

    1. Not this one. You listed HH, HT and TT. HT and TH are different outcomes, so there are 4 outcomes and 2 of them have one of each. Go to M4.6.1. Try another.

    2. Yes, that is right. The 4 outcomes are HH, HT, TH and TT, all equally likely. HT and TH have one of each, so the probability is 2 out of 4, which is 1/2.

    3. Not this one. That is the chance of one particular outcome, such as HT. One head and one tail can happen two ways, HT and TH. Go to M4.6.1. Try another.

    See the answer

    1/2

    The 4 outcomes are HH, HT, TH and TT, all equally likely. HT and TH have one of each, so the probability is 2 out of 4, which is 1/2.

    • If you chose A: You listed HH, HT and TT. HT and TH are different outcomes, so there are 4 outcomes and 2 of them have one of each. Go to M4.6.1.
    • If you chose C: That is the chance of one particular outcome, such as HT. One head and one tail can happen two ways, HT and TH. Go to M4.6.1.

  6. In the table of two dice, a total of 7 fills 6 cells and a total of 12 fills 1 cell. How many times as likely is a total of 7?

    Tap your answer.

    1. Not this one. You used the total, 12. Compare the numbers of ways: 6 ways against 1 way. Go to M4.6.1. Try another.

    2. Not this one. You took away, 6 − 1. "Times as likely" compares by dividing. Go to M4.6.1. Try another.

    3. Yes, that is right. 6 cells for a total of 7 and 1 cell for a total of 12. 6 ÷ 1 = 6, so a total of 7 is 6 times as likely.

    See the answer

    6 times as likely

    6 cells for a total of 7 and 1 cell for a total of 12. 6 ÷ 1 = 6, so a total of 7 is 6 times as likely.

    • If you chose A: You used the total, 12. Compare the numbers of ways: 6 ways against 1 way. Go to M4.6.1.
    • If you chose B: You took away, 6 − 1. "Times as likely" compares by dividing. Go to M4.6.1.

  7. Two dice are rolled and added. What is the probability that the total is 9?

    See the answer

    1/9

    A total of 9 is 3 and 6, 4 and 5, 5 and 4, 6 and 3. That is 4 of 36 cells, and 4/36 = 1/9.

  8. A coin is tossed and a dice is rolled. What is the probability of a tail and a number greater than 4?

    See the answer

    1/6

    The outcomes are T5 and T6. That is 2 of the 12 equally likely outcomes, and 2/12 = 1/6.

Keep it

Copy this into your exercise book.

List every outcome in a table. Check that they are equally likely. P(event) = outcomes in the event ÷ outcomes in all. Two dice make 36 pairs, but the totals are not equally likely.

Draw: A 6 by 6 table of two dice with the cells that give a total of 7 hatched.

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.