S2 January 2006 Q1
1. A fair coin is tossed 4 times.
Find the probability that
(a) an equal number of head and tails occur (2)
(b) all the outcomes are the same, (3)
(c) the first tail occurs on the third throw. (2)
| Scheme | Marks |
|---|---|
| Let \(X\) be the random variable the number of heads. \(X \sim \mathrm{Bin}(4, 0.5)\) | |
| \(\mathrm{P}(X = 2) = {}^4\mathrm{C}_2\,0.5^2\,0.5^2\) | M1 |
| \(= 0.375\) | A1 |
| (2) |
Notes
M1 use of Binomial including \({}^n\mathrm{C}_r\)
A1 or equivalent
1a) 2,4,6 acceptable as use of binomial.
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X = 4)\) or \(\mathrm{P}(X = 0)\) | B1 |
| \(= 2 \times 0.5^4\) | M1 |
| \(= 0.125\) | A1 |
| (3) |
Notes
M1 \((0.5)^4\)
A1 or equivalent
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(HHT) = 0.5^3\) | M1 |
| \(= 0.125\) | A1 |
| or \(\mathrm{P}(HHTT) + \mathrm{P}(HHTH)\) \(= 2 \times 0.5^4\) \(= 0.125\) | |
| (2) | |
| (7 marks) |
Notes
M1 no \({}^n\mathrm{C}_r\)
A1 or equivalent