Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8: 2/3 × 3/4 = ?
Answer
1/2. Multiply the tops (6) and the bottoms (12), then simplify. - From Lesson 3: a bag has 12 counters and 5 are green. What is P(not green)?
Answer
7/12. 12 − 5 = 7 are not green. - From Year 7: 0.4 × 0.5 = ?
Answer
0.2. 4 × 5 = 20, and two decimal places in all gives 0.20.
Words used in this lesson
- tree diagram
- a diagram that shows every route through several stages, with a probability on each branch. For example: the picture for two draws from a drum.
- branch
- one line of a tree diagram, labelled with a probability. For example: the line marked 3/7.
- route
- a path from the start to the end of a tree; one possible outcome. For example: red then blue.
- with replacement
- each item is put back before the next draw, so the stages are independent. For example: draws from a bag where the sweet is returned.
- without replacement
- an item is kept out, so the next stage has changed. For example: names pulled from a hat and set aside.
- independent
- the result of one stage does not change the probabilities of the next. For example: a coin toss and a die roll.
The drum at the school fair
At a school fair in Cusco a drum holds 100 tickets and some of them win a prize. A student draws one ticket and keeps it, then draws a second ticket. Their friend does the same, but puts the first ticket back before drawing again.
Predict first: if the first ticket was a winner, is the chance that the second ticket is a winner the same for both students? Who is more likely to win two prizes, and why? Use the first figure to test your ideas with counters, then move to the second figure to see how the chance of at least one success grows with more trials, and whether it ever reaches 100%.
With replacement, 4 red and 3 blue: the second draw is from 7 counters, so P(same colour) = (16 + 9)/49 = 25/49.
Show me the steps
- With replacement, 4 red and 3 blue: each draw is from 7 counters, so every route has 49 on the bottom, and P(same colour) = (16 + 9)/49 = 25/49.
- Press Without replacement: after one counter is kept out the second draw is from 6 counters, so the second-stage fractions change and every route has 42 on the bottom: P(same colour) = (12 + 6)/42 = 18/42 = 3/7.
- Without replacement, set red to 1 and blue to 3: after the single red counter is drawn there are no reds left, so the RR route is 0/12 and P(same colour) = (0 + 6)/12 = 6/12 = 1/2.
- Press With replacement again: the second stage repeats the first stage, because nothing was removed, so the routes are independent of each other.
Multiply along a route, add the routes that count.
- Each stage is a set of branches, and the probabilities on the branches from one point always add to 1.
- The probability of a whole route is the product of the probabilities on its branches (“and then”).
- If an event can happen along several routes, add their probabilities (“or”).
- With replacement the second stage repeats the first, so the events are independent. Without replacement the numbers on the second stage change, because the bag has one fewer counter.
“At least one” is usually quicker as 1 − P(none).
Chance of no success twice: 70% × 70% = 49%, so P(at least one success) = 100% − 49% = 51%.
Show me the steps
- 30% each time and 2 trials: no success is 70% × 70% = 49%, so at least one success is 100% − 49% = 51%.
- Set 4 trials with 30%: no success is 70% × 70% × 70% × 70% = 24.01%, so at least one is 75.99%. More trials give more chances.
- Set 10% and 2 trials: no success is 90% × 90% = 81%, so at least one is only 19%.
- Set 90% and 2 trials: no success is 10% × 10% = 1%, so at least one is 99%.
Multiply along a branch; do not add. Two draws that both have to happen are an “and”, so the chances are multiplied. Adding along a branch gives a number that can be bigger than 1. And without replacement, remember that the bottom number goes down by 1 on the second stage as well as the top number, if the first counter was the same colour.
Worked example: one of each, without replacement
A bag holds 6 green and 2 yellow sweets. Two are taken without replacement. Find P(one of each colour).
- First draw: P(green) = 6/8 and P(yellow) = 2/8.
- Green then yellow: 6/8 × 2/7 = 12/56. (After a green is taken, 7 remain, and 2 of them are yellow.)
- Yellow then green: 2/8 × 6/7 = 12/56.
- Add the two routes: 12/56 + 12/56 = 24/56 = 3/7.
Your go. The first step is given. A bag holds 7 grey and 3 white counters. Two are taken with replacement. Find P(exactly one grey).
- There are two routes: grey then white, and white then grey. Because each counter is put back, each draw has P(grey) = 7/10 and P(white) = 3/10.
- Multiply along each route, then add the two routes.
Check the answer
Grey then white: 7/10 × 3/10 = 21/100, and white then grey is also 21/100. Together P = 42/100 = 21/50. A common slip is to find only one route and give 21/100.
With a GDC (optional)
A GDC can multiply fractions exactly. Use the fraction template (often an a b/c key or an entry in the maths menu) and type 3/8 × 2/7: it gives 3/28 rather than a decimal. Use it to check the numerators and denominators you wrote on a tree.
It will also turn a fraction into a decimal. A written answer should stay as an exact fraction unless the question asks for a decimal, and a rounded decimal should say how it was rounded. Everything here can be done by hand.
Check before you go on
1. A bag holds 4 red and 6 green counters. One is taken and not replaced. It is green. What probability goes on the branch “second counter is green”?
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 goalkeeper saves a penalty with probability 0.35. Two penalties are taken independently. What is the probability that both are saved?
3. A tin holds 6 chocolate and 4 plain biscuits. Two are taken without replacement. Written over 90, the probability that both are chocolate is ?/90. What is the missing top number?
Level 2 · Building
4. A drawer holds 7 black and 4 white socks. Two are taken without replacement. Written over 110, the probability of a matching pair is ?/110. What is the missing top number?
5. Each of 2 independent seeds germinates with probability 0.85. What is the probability that at least one germinates?
6. A bag holds 8 red and 3 green counters. Two are taken without replacement. Written over 110, the probability that they are different colours is ?/110. What is the missing top number?
Level 3 · Stretch
7. A bag holds 8 red and 4 blue counters. Three are taken without replacement. Written over 1320, the probability that all three are red is ?/1320. What is the missing top number?
8. A bag holds 7 red and 3 blue counters. Two are taken without replacement. Written over 90, the probability that the second counter is red is ?/90. What is the missing top number?
Something to think about
A bag holds red and blue counters. Two are drawn without replacement. Is P(first red, second blue) always the same as P(first blue, second red)? Try some numbers, then explain why, in a sentence, with a small example.
One way to think about it
They are equal. Both routes use the same two draws: one red and one blue. Red then blue is r/n × b/(n − 1), and blue then red is b/n × r/(n − 1). The same four numbers are multiplied in a different order, so the products match. That is why a two-stage question about one of each can be answered by doubling one route.
Exit ticket
9. A hat holds the names of 6 girls and 3 boys. Two names are drawn without replacement. What is the probability that both are girls?