4400 Higher November 2007 Paper 3 Q17
17 Each time Nikos has a shot at goal, the probability that he will score a goal is \(\dfrac{3}{4}\)
Nikos takes two shots.
(a) Complete the probability tree diagram.
(b) Calculate the probability that Nikos will score
(i) two goals, (2)
(ii) exactly one goal. (3)
Nikos now takes another three shots.
(c) Calculate the probability that he will score exactly 1 goal or exactly 2 goals. (3)
| Scheme | Marks |
|---|---|
| All correct | B2 |
| (2) |
Notes
B2: ignore branches for 3rd shot
correct structure & labels or probs: B1
| Scheme | Marks |
|---|---|
| (i) \((\frac{3}{4})^2\) | M1 |
| \(\frac{9}{16}\) or 0.5625 | A1 |
| (ii) \(\frac{3}{4} \times \frac{1}{4}\) | M1 |
| \(\frac{3}{4} \times \frac{1}{4} + \frac{1}{4} \times \frac{3}{4}\) | M1 |
| \(\frac{3}{8}\) or \(\frac{6}{16}\) or 0.375 | A1 |
| (5) |
Notes
A1: or 0.563
| Scheme | Marks |
|---|---|
| \((\frac{3}{4})^3\) or \((\frac{1}{4})^3\) | M1 |
| \(1 - ((\frac{3}{4})^3 + (\frac{1}{4})^3)\) | M1 |
| \(\frac{9}{16}\) or 0.5625 | A1 |
| (3) | |
| (10 marks) |
Notes
M1: \((\frac{3}{4})^2 \times (\frac{1}{4})\) or \((\frac{1}{4})^2 \times \frac{3}{4}\)
M1: \(3 \times (\frac{3}{4})^2 \times (\frac{1}{4}) + 3 \times (\frac{1}{4})^2 \times \frac{3}{4}\)
A1: or 0.563
\((\frac{3}{4})^2 \times (\frac{1}{4})^3\) or \((\frac{1}{4})^4 \times \frac{3}{4}\) M1
\(10 \times (\frac{3}{4})^2 \times (\frac{1}{4})^3 + 5 \times (\frac{1}{4})^4 \times \frac{3}{4}\) M1
\(\frac{105}{1024}\) A1