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Listing Equally Likely Outcomes

Maths · Stage 3
Year 5
Lesson Guide
Explanation
Write every outcome out in a fixed order and none gets missed. Two coins give four: head-head, head-tail, tail-head, tail-tail. Head-then-tail and tail-then-head are genuinely different, which is why there are four and not three, and why one head and one tail covers two of the four — a chance of one half. Listing all the head-first outcomes and then all the tail-first ones keeps the list complete.
Your Turn
What is the chance of two heads when two coins are tossed? Check: 1 out of 4.
Watch Out
HT and TH are different outcomes. Two coins have four outcomes, not three.
Explanation
Counting ways over the total only works when every outcome is equally likely. On a spinner with four equal sections each colour is one out of four. But on a spinner with three colours where blue covers half the circle, the chance of blue is not one out of three — it is a half. So check the sections are the same size before you count them, and when they are not, count by area rather than by colour.
Your Turn
A spinner has 3 colours but blue covers half of it. What is the chance of blue? Check: 1 out of 2.
Watch Out
Three colours does not mean one out of three. Look at how much space each one takes.
Explanation
One ordinary die gives six equally likely outcomes. The even numbers are two, four and six, three out of six, a half. The numbers more than four are five and six, two out of six. Even and more than four is only the six, one out of six, because both conditions must hold; even or more than four collects two, four, five and six, four out of six. And is narrower, or is wider. Write down the numbers that qualify and count them rather than doing it in your head.
Your Turn
What is the chance of rolling a number more than 4? Check: 5 and 6, so 2 out of 6.
Watch Out
"And" needs both conditions at once; "or" needs only one. They give different counts.
Explanation
When two things happen at once, a table lists every pairing. A coin and a die give two times six, twelve outcomes. The chance of a head and a six is just one of those, one out of twelve; the chance of a head and an even number is H two, H four and H six, three out of twelve, which is one out of four. Multiplying gives the total quickly, and the table is there to count the ones that qualify.
Your Turn
How many outcomes are there when a coin and a die are used together? Check: 2 × 6 = 12.
Watch Out
Multiply to get the TOTAL number of outcomes; do not multiply the qualifying counts.
Explanation
Three steps every time: list every outcome, mark the ones that qualify, and count both numbers. Writing the list takes about ten seconds and removes the guesswork entirely — most errors in this chapter come from trying to hold the outcomes in your head and missing one. Quiz and exercises now.
Your Turn
List the outcomes for two coins. Check: HH, HT, TH, TT.
Watch Out
Answer in the form asked for: "2 out of 4" and "a half" are the same, but read which one is wanted.
📌 Key Points — this lesson in a nutshell
  • Write out every outcome in a fixed order, then count the ones the question wants.
  • Two coins give four outcomes: HH, HT, TH, TT. Head-then-tail and tail-then-head are different.
  • Outcomes are equally likely only when every one is as easy as every other — four equal spinner sections, six faces of a fair die.
  • The probability is then how many count, over how many there are.
Step 1: Introduction
Step 1 of 5
Tossing two coins HHHTTHTT first coin, then second coin — four outcomes HH, HT, TH, TT — four outcomes, and each is as likely as the others. HT and TH are different: in one the first coin is a head, in the other the second is. Listing them in a fixed order — all the H-first ones, then all the T-first ones — makes sure none is missed. "One head and one tail" covers TWO of the four outcomes, so its chance is 2 out of 4, a half. Thinking there are three outcomes — two heads, two tails, one of each — is the classic mistake.
Four equal sections, or not equal sections: each 1 out of 4 unequal: blue is 1 out of 2 "Ways over total" only works when every outcome is equally likely. On the second spinner there are 3 colours, but the chance of blue is not 1 out of 3 — blue covers half the circle. Check the sections are the same size before counting them. Count area, not colours.
Rolling one ordinary die Six equally likely outcomes: 1, 2, 3, 4, 5, 6. Even numbers: 2, 4, 6 — that is 3 out of 6, a half. More than 4: 5 and 6 — 2 out of 6, which is 1 out of 3. Even AND more than 4: only 6 — 1 out of 6. Both conditions must hold. Even OR more than 4: 2, 4, 5, 6 — 4 out of 6. "Or" collects; "and" narrows. Write the numbers that qualify, then count them. Do not try to do it in your head.
A coin and a die together 123456HH1H2H3H4H5H6TT1T2T3T4T5T6 A table lists every pairing: 2 coin results × 6 die results = 12 outcomes. Chance of a head and a 6: just H6, so 1 out of 12. Chance of a head and an even number: H2, H4, H6 — 3 out of 12, which is 1 out of 4. Two coins the same way: 2 × 2 = 4 outcomes, which matches the list HH, HT, TH, TT. Multiply the numbers of outcomes to get the total; use the table to count the ones that qualify.
Reading the question How many outcomes when two coins are tossed?4 Chance of one head and one tail?2 out of 4 Chance of an even number on a die?3 out of 6 Chance of even AND more than 4?1 out of 6 List every outcome · mark the ones that qualify · count both numbers. Writing the list takes ten seconds and removes the guesswork. Quiz and exercises now.
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