GCSE Maths · Higher · Chapter 19: Combined events · Lesson 2

Tree diagrams and independent events

Grades 5–8About 50 minutesNo calculator40 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out $\frac{5}{8} \div \frac{1}{4}$ .
[1 mark]
Answer
A2
From chapter 17
Substitute $y = 2 x$ into $x^{2} + y^{2} = 5$ and find the positive value of $x$ .
[1 mark]
Answer
A3
Last lesson
Events $A$ and $B$ cannot both happen. P(A) $= 0.3$ , P(B) $= 0.45$ . Find P(A or B).
[1 mark]
Answer
A4
From chapter 10
Find the equation of the line through $(2, 4)$ and $(5, 13)$ .
[1 mark]
Answer

Example, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it.

Example 1

The probability that it rains on any day is $0.25$ , and the days are independent. Work out the probability that it rains on at least one of two days.

Hint The only way it does not happen is no rain on both days:

0.75 \times 0.75 = 0.5625

. Take that from

1

B1
A house has two smoke alarms. Each one fails with probability $0.15$ , independently of the other. Work out the probability that at least one alarm fails.
[2 marks]
Answer

Example 2

A drawer holds $6$ black and $3$ white socks. Two are taken at random without replacement. Work out the probability that both are black.

Hint After one black sock has gone there are

5

black socks left out of

8

B2
A box holds $7$ white and $5$ brown eggs. Two are taken at random without replacement. Work out the probability that both are white.
[2 marks]
Answer

Practice

Grades 6–7

C1
A fair dice is thrown three times. Work out the probability that the first six comes on the third throw.
[2 marks]
Answer
C2
A bag holds $5$ red and $4$ white beads. Two are taken at random without replacement. Work out $P (at least one is red)$ .
[3 marks]
Answer

C3
A coin is biased. It is thrown twice, and the probability of two heads is $0.4225$ . Work out the probability of two tails.
[3 marks]
Answer
C4
$P (A) = 0.6$ , $P (B) = 0.35$ and $P (A and B) = 0.21$ . Are $A$ and $B$ independent? Show how you decide.
[2 marks]
C5
Two spinners are spun. The tree diagram shows the probability that each lands on red. Complete the tree diagram, then work out the probability that exactly one of them lands on red.
[3 marks]
Answer
C6
A spinner lands on $1$ , $2$ or $3$ . $P (1) = 0.4$ , $P (2) = 0.5$ and $P (3) = 0.1$ . It is spun twice and the two scores are added. Work out the probability that the total is $4$ .
[3 marks]
Answer

Exam-style questions

Grades 7–8

D1
A bag holds $8$ counters. $3$ are blue and the rest are yellow. Two counters are taken at random without replacement. Sam says, "The probability that both are blue is $\frac{3}{8} \times \frac{3}{8} = \frac{9}{64}$ ." Explain what Sam has done wrong, and work out the correct probability.
[3 marks]
Answer

D2
A box holds $12$ raffle tickets, and $4$ of them win a prize. Ana takes a ticket, puts it back, then takes another. Ben takes two tickets without putting the first back. Who is more likely to get at least one winning ticket? You must show your working.
[4 marks]
D3
The probability that Ella is late for school is $0.15$ . If she is late, the probability that she gets a detention is $0.3$ . If she is on time, the probability that she gets a detention is $0.05$ .
(a)
Complete the tree diagram. Work out the probability that Ella is late and gets a detention.
[2 marks]
Answer
(b)
Work out the probability that Ella gets a detention.
[2 marks]
Answer
(c)
Ella goes to school on $400$ days. On how many of them would you expect her to get a detention?
[1 mark]
Answer
Total for D3: 5 marks

Extension

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A pot holds $3$ red, $4$ blue and $5$ green pens. Three pens are taken at random without replacement. Work out the probability that there is one pen of each colour.
[4 marks]
Hint 1 · What to try
Work out the probability of red, then blue, then green, in that order.
Hint 2 · The first line
$\frac{3}{12} \times \frac{4}{11} \times \frac{5}{10} = \frac{60}{1320}$ . Every order of the three colours gives the same answer.
Hint 3 · The full method
There are $6$ orders, so multiply by $6$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can multiply along a tree and add between its paths.
I can use $1 - P (none)$ for "at least one".
I can change the second probability when nothing is put back.

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