June 2023 Paper 2 Q4
4 A biased octagonal dice has faces numbered from 1 to 8. The discrete random variable \(X\) is the score obtained when the dice is rolled once. The probability distribution of \(X\) is shown in the table below.
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| \(\mathrm{P}(X = x)\) | \(p\) | \(p\) | \(p\) | \(p\) | \(p\) | \(p\) | \(p\) | \(3p\) |
(a) Determine the value of \(p\). [2]
(b) Find the probability that a score of at least 4 is obtained when the dice is rolled once. [1]
The dice is rolled 30 times.
(c) Determine the probability that a score of 8 occurs exactly twice. [2]
| Scheme | Marks | AO |
|---|---|---|
| \(7p + 3p = 1\) oe | M1 | 1.2 |
| \(p = 0.1\) or \(\frac{1}{10}\) isw cao | A1 | 1.1 |
| [2] |
Notes
M1: must see some reasoning; allow explanation in words
A1: if unsupported, allow SC1 for correct answer
| Scheme | Marks | AO |
|---|---|---|
| 0.7 or \(\frac{7}{10}\) | B1FT | 1.1 |
| [1] |
Notes
B1FT: their \(7p\)
| Scheme | Marks | AO |
|---|---|---|
| B(30, their 0.3) seen or used | M1 | 1.1 |
| awrt 0.0018 isw | A1 | 1.1 |
| [2] |
Notes
M1: FT their \(3p\); allow M1 for \((\textit{their }0.3)^2 \times (\textit{their }0.7)^{28}\)
A1: not from wrong working; if unsupported, allow SC1 for correct answer