The probability that Freya will pass the exam is 0.8 The probability that Ria will pass the exam is 0.85
(a) Complete the probability tree diagram.(2)
(b) Work out the probability that only one of Freya or Ria will pass the exam. (3)
Mark scheme (a)
Answer
Mark
Mark scheme
0.2, 0.15, 0.85, 0.15
B2
all placed correctly
(B1
for 2 or 3 placed correctly)
Mark scheme (b)
Answer
Mark
Mark scheme
0.29
M1
for one correct product, ft their tree diagram eg \(0.8 \times 0.85\ (= 0.68)\) or \(0.8 \times \text{``}0.15\text{''}\ (= 0.12)\) or \(\text{``}0.2\text{''} \times \text{``}0.85\text{''}\ (= 0.17)\) or \(\text{``}0.2\text{''} \times \text{``}0.15\text{''}\ (= 0.03)\)
M1
for a complete method, ft their tree diagram eg \(\text{``}0.2\text{''} \times \text{``}0.85\text{''} + 0.8 \times \text{``}0.15\text{''}\) or \(1 - (0.8 \times 0.85 + \text{``}0.2\text{''} \times \text{``}0.15\text{''})\)
A1
for 0.29 oe or ft their tree diagram
Additional guidance
Working for part (b) may be seen in part (a) To apply ft values must be <1
At the start of the maze you can turn left or right.
45 children turn left.
75 people in total turn left.
(a) Complete the frequency tree. [4 marks]
(b) What fraction of the children turn left?
Give your answer in its simplest form. [2 marks]
Mark scheme (a)
Answer
Mark
Comments
80 children and 40 adults
B1
45 left (children) and 35 right (children)
B1
30 left (adults)
B1ft
ft \(75 -\) their 45, where their 45 is an integer \(\geqslant 0\) and answer an integer \(\geqslant 0\)
10 right (adults)
B1ft
ft their \(40 -\) their 30, where both integers are \(\geqslant 0\) and answer an integer \(\geqslant 0\)
Additional guidance
Use of relative frequency or probability for an answer is B0 for that answer
Ignore working outside the frequency tree
Mark scheme (b)
Answer
Mark
Comments
\(\dfrac{9}{16}\)
B2ft
ft their 45 and/or ft their 80 B1ft \(\dfrac{\text{their } 45}{\text{their } 80}\) not fully simplified or correct simplification of their fraction, using numbers from their tree
Additional guidance
B2ft can only be awarded if their numbers can be simplified, otherwise B1ft
Do not ignore further work for B2 after correct answer seen \(\dfrac{9}{16} = \dfrac{3}{4}\)
6850 of the fans support the Home team. The remaining fans support the Away team. 20% of the Home fans wear a scarf. 2319 of all the fans wear a scarf.
Complete the frequency tree. [5 marks]
Mark scheme
Answer
Mark
Comments
1550 in Away
B1
1370 in Home Yes
B1
5480 in Home No
B1ft
ft 6850 − their 1370 their 1370 must be less than 6850
949 in Away Yes
B1ft
ft 2319 − their 1370 their 1370 must be less than 2319
601 in Away No
B1ft
ft their 1550 − their 949 their 1550 must be greater than their 949
Additional guidance
If Away oval is blank then condone an indication of 1550 as Away
If Home Yes oval is blank then condone an indication of 1370 as Home Yes
16 A fair spinner has five equal sections numbered 1, 2, 3, 4 and 5
A fair six-sided dice has five red faces and one green face.
The spinner is spun.
If the spinner shows an even number, the dice is thrown.
(a) Complete the tree diagram for the spinner and the dice. [2 marks]
(b) Work out the probability of getting an even number and the colour green. [2 marks]
Mark scheme (a)
Answer
Mark
Comments
\(\dfrac{2}{5}\) Even and \(\dfrac{3}{5}\) Odd
B1
oe fractions, decimals or percentages
Two branches from Even labelled Red \(\dfrac{5}{6}\) \(\qquad\) Green \(\dfrac{1}{6}\)
B1
oe fractions, decimals or percentages Branches from Odd is B0 Allow equivalent labelling eg R and G Green and Not Green
Additional guidance
In decimals, allow for \(\dfrac{5}{6}\) and \(\dfrac{1}{6}\) 0.83 and 0.17 or 0.833 and 0.167 or 0.834 and 0.166 or 0.84 and 0.16 or better truncation or rounding (sum of pair must equal 1) In percentages, allow for \(\dfrac{5}{6}\) and \(\dfrac{1}{6}\) 83% and 17% or 83.3% and 16.7% or 83.4% and 16.6% or 84% and 16% or better truncation or rounding (sum of pair must equal 100%)
Ignore any attempts to combine probabilities to the right of the tree diagram
Mark scheme (b)
Answer
Mark
Comments
their \(\dfrac{2}{5} \times\) their \(\dfrac{1}{6}\)
M1
their P(Even) \(\times\) their P(Green) ft from (a) if \(0 \lt\) both probabilities \(\lt 1\)
\(\dfrac{2}{30}\) or \(\dfrac{1}{15}\)
A1ft
oe fraction or decimal ft from (a) if \(0 \lt\) both probabilities \(\lt 1\)
Additional guidance
Allow 0.06 or 6% or better truncation or rounding or 0.07 or 7% for \(\dfrac{2}{30}\)
If the dice branches are not labelled there is no ft from (a)
If (a) has no attempt or an incorrect answer full marks can still be gained here for correct working (and answer)
Ignore further attempts to simplify or convert to a decimal or percentage after a correct fraction is seen eg \(\;\dfrac{2}{30} = \dfrac{1}{10}\) or \(\dfrac{4}{60} = 0.165\)
20% of the people who say Yes drink at least three cups each day.
(a) Complete the frequency tree. [4 marks]
(b) What fraction of the 500 people drink at least three cups of coffee each day?
Give your answer in its simplest form. [2 marks]
Mark scheme (a)
Answer
Mark
Comments
450 in Drink coffee Yes
B1
50 in Drink coffee No
B1ft
ft 500 \(-\) their 450
90 in At least three cups Yes
B1ft
ft their 450 \(\div\) 5
360 in At least three cups No
B1ft
ft their 450 \(-\) their 90
Additional guidance
for 90 ft, their 450 ÷ 5 must be truncated or rounded up to the nearest whole number
for 360 ft, their 450 \(-\) their 90 must give a positive integer
Accept unambiguous values elsewhere but diagram values take precedence
Correct relative frequencies seen, withhold first B1 that would have been awarded. eg \(\dfrac{400}{500}, \dfrac{100}{500}, \dfrac{80}{400}, \dfrac{320}{400}\) eg \(\dfrac{400}{500}, \dfrac{100}{500}, \dfrac{80}{500}, \dfrac{320}{500}\)
B0 B0ft B1ft B1ft B0 B0ft B0ft B0ft
Do not accept probabilities eg \(\dfrac{9}{10}, \dfrac{1}{10}, \dfrac{4}{5}, \dfrac{1}{5}\) eg 0.9, 0.1, 0.8, 0.2
