Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Write 3/5 as a decimal and as a percentage.
Answer
0.6 and 60%. 3/5 = 6/10 = 60/100. - Which is greater, 0.4 or 3/10?
Answer
0.4. 3/10 is 0.3, and 0.4 is four tenths. - Simplify 12/20.
Answer
3/5. Divide the top and the bottom by 4.
Words used in this lesson
- probability
- a number from 0 to 1 that says how likely an event is. For example: the probability of a red counter is 0.3.
- impossible
- cannot happen, probability 0. For example: rolling 7 on a standard die.
- certain
- must happen, probability 1. For example: rolling a number below 7.
- even chance
- equally likely to happen or not, probability 1/2. For example: a fair coin landing on heads.
- event
- a result or set of results you are asking about. For example: picking a red counter.
From impossible to certain
A bag holds 20 counters and 5 of them are red. You pick one without looking. Before you open the figure, draw a line from 0 to 1 on paper and mark where “you pick a red one” belongs. Is it nearer 0, nearer 1, or in the middle?
Probability puts every chance on that one line. A probability of 0 means the event cannot happen, so it is impossible. A probability of 1 means it must happen, so it is certain. Everything else lies strictly between.
A probability is a number from 0 to 1. 0 is impossible and 1 is certain. The larger the number, the more likely the event.
One position, three names. For a bag of counters, each as likely to be picked as any other, P(red) is the red counters out of all the counters. 3/20, 0.15 and 15% are the same place on the scale.
Slide the red counters and watch the marker. Then set the bag to 5 red, and compare it with the line you drew.
12 red counters out of 20: a probability of 12/20 = 0.6 = 60%, between even chance and certain, so it is likely.
Show me the steps
- 0 red counters: 0/20 = 0 = 0%. A red cannot be picked: impossible.
- 10 red counters: 10/20 = 0.5 = 50%. Half the counters are red: an even chance.
- 4 red counters: 4/20 = 0.2 = 20%. Less than half: unlikely.
- 17 red counters: 17/20 = 0.85 = 85%. More than half: likely.
- 20 red counters: 20/20 = 1 = 100%. Every counter is red: certain.
Two outcomes does not mean 50%. Picking a counter, you either get a red or you do not, but that does not make each half of the chance. With 1 red counter in a bag of 20, red is 1/20 = 0.05, close to the impossible end of the scale. Half is only the right place when half the counters are red.
Worked examples
A spinner has 8 equal sections and 6 are shaded. Write the probability of landing on shaded as a fraction, a decimal and a percentage.
- Shaded sections out of all sections: 6/8, which simplifies to 3/4.
- 3 ÷ 4 = 0.75, and 0.75 × 100 = 75%.
- So P(shaded) = 3/4 = 0.75 = 75%: past the middle of the scale, so it is likely.
A standard die has faces 1 to 6. Where do “shows 7” and “shows a number below 7” sit?
- No face shows 7, so 0 of the 6 faces work: P = 0/6 = 0, impossible.
- Every face is below 7, so 6 of the 6 faces work: P = 6/6 = 1, certain.
Your go. The first step is given. A bag holds 25 counters and 10 are green. Write P(green) as a decimal and as a percentage.
- Green counters out of all counters: 10/25.
- Finish it, then check.
Check the answer
10/25 = 2/5 = 0.4, which is 40%.
Check before you go on
1. Which of these cannot be a probability?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A bag holds 10 counters and 7 are red. Write the probability of picking a red counter as a decimal.
3. A raffle sells 50 tickets and Amina holds 10 of them. Write the probability that Amina wins as a percentage. Type the number only.
Level 2 · Building
4. A bag holds 40 counters and 12 are blue. Write P(blue) as a decimal.
5. A forecast gives a 35% chance of rain tomorrow. Write the probability that it does not rain as a decimal.
6. A standard die is rolled. Which event has a probability of 0?
Level 3 · Stretch
7. A bag holds 4 red counters and 6 blue counters. How many green counters must be added so that the probability of picking green is 0.5?
8. Event A has probability 0.18. Event B is three times as likely as A. Find P(B).
Something to think about
Put 20 counters in a bag of red, blue and green counters so that picking red is just as likely as picking blue, but picking green is impossible. How many counters of each colour could the bag hold? Is there more than one answer?
One way to think about it
The bag holds only red, blue and green counters, so red and blue must each be half of it: 10 red, 10 blue and 0 green is the only way with 20 counters. With a bigger bag there are still only two colours present, each half of it, so with a different number of counters, red and blue are still half each: 6 and 6, 50 and 50, and so on, any even number.
Exit ticket
9. Which list puts 3/5, 30% and 0.4 in order from least likely to most likely?