Foundation statistics and probability clinic

(a) $\{2, 4\}$

1. In both $A$ and $B$: 2 and 4.

• B1 Correct answer: 2 and 4.

(b) $\frac{2}{10}$

1. $A \cup B = \{1, 2, 3, 4, 5, 6, 8, 10\}$: 8 numbers.
2. $(A \cup B)'$ is the rest: $\{7, 9\}$, so $P = \frac{2}{10}$.

• M1 Identifying $\{7, 9\}$ (or $A \cup B$ with 8 members).
• A1 $\frac{2}{10}$ or equivalent.

(a) Positive

1. Higher temperatures go with more sales.

• B1 Correct answer: Positive.

(b) An unusual day, such as the shop opening late or a rainstorm: it is an outlier.

1. The point does not fit the pattern, so something else affected sales that day.

• C1 A sensible reason that the day is unusual (an outlier), such as rain or the shop being shut for part of the day.

(a) $34$

1. Use inclusion-exclusion to count the union once.
   $$36+26-16$$
2. Subtract the union from the group total.
   $$80-46$$
3. Therefore $34$.

• M1 Use inclusion-exclusion to count the union once.
• M1 Subtract the union from the group total.
• A1 Correct answer: $34$

(b) $3/8$

1. Remove the overlap from each drink total.
   $$\frac{36-16+26-16}{80}$$
2. Therefore $3/8$.

• M1 Remove the overlap from each drink total.
• A1 Correct answer: $3/8$

(a) $4/25$

1. Use the same red fraction on both draws.
   $$4/10\times 4/10$$
2. Therefore $4/25$.

• M1 Use the same red fraction on both draws.
• A1 Correct answer: $4/25$

(b) $21/25$

1. Both red is the only way to have no blue.
   $$1-(4/25)$$
2. Therefore $21/25$.

• M1 Both red is the only way to have no blue.
• A1 Correct answer: $21/25$

(a) $(15,15)$

1. Use the midpoint of the second class for the horizontal coordinate.
   $$\frac{10+20}{2}$$
2. Therefore $(15,15)$.

• M1 Use the midpoint of the second class for the horizontal coordinate.
• A1 Correct answer: $(15,15)$

(b) $38$

1. Add all class frequencies.
   $$12+15+11$$
2. Therefore $38$.

• M1 Add all class frequencies.
• A1 Correct answer: $38$

(a) $96$

1. Find the gradient from two points on the line.
   $$\frac{135-31}{8}$$
2. Move five hours along the line from x = 2.
   $$31+5\times 13$$
3. Therefore $96$.

• M1 Find the gradient from two points on the line.
• M1 Move five hours along the line from x = 2.
• A1 Correct answer: $96$

(b) 25 hours lies outside the observed range, so this is extrapolation; the relationship may change beyond the recorded data.

1. Therefore 25 hours lies outside the observed range, so this is extrapolation; the relationship may change beyond the recorded data.

• C1 Correct conclusion with supporting reasoning: 25 hours lies outside the observed range, so this is extrapolation; the relationship may change beyond the recorded data.

(a) $255/512$

1. Count the red-blue route.
   $$15/32\times 17/32$$
2. Include the blue-red route, which has the same probability.
   $$2\times 15\times 17/32^{2}$$
3. Therefore $255/512$.

• P1 Count the red-blue route.
• P1 Include the blue-red route, which has the same probability.
• A1 Correct answer: $255/512$

(b) $49.8$

1. Multiply the probability of one mixed pair by 100 trials.
   $$100\times 2\times 15\times 17/32^{2}$$
2. Therefore $49.8$.

• P1 Multiply the probability of one mixed pair by 100 trials.
• A1 Correct answer: $49.8$

(a) 0.1

1. Independent: multiply. $0.4 \times 0.25 = 0.1$.

• M1 $0.4 \times 0.25$.
• A1 Correct answer: 0.1.

(b) £40

1. Expected winners: $0.1 \times 200 = 20$.
2. Money in minus money out.
   $$200 - 20 \times 8 = 200 - 160 = 40$$

• P1 Expected winners, 20.
• P1 Money out, $20 \times 8 = 160$.
• A1 Correct answer: £40.