| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Multiple independent binomial calculations |
| Difficulty | Moderate -0.3 This question involves straightforward applications of binomial probability formulas and basic probability laws. Part (i) is a direct binomial calculation, part (ii) uses the law of total probability with given conditional probabilities, and part (iii) applies the variance formula for a binomial distribution. All three parts are routine textbook exercises requiring recall and standard methods rather than problem-solving or insight. |
| Spec | 2.03d Calculate conditional probability: from first principles5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P(> 5) = ^7C_6(0.6)^6(0.4) + (0.6)^7 = 0.1306 + 0.02799 = 0.159\) | M1 | Summing 2 or 3 binomial probs of the form \(^nC_r(0.6)^r(0.4)^{n-r}\) |
| A1 | Correct answer | |
| (ii) \(P(\text{bark}) = P(\text{park, bark}) + P(\text{not park, bark}) = 0.6 \times 0.35 + 0.4 \times 0.75 = 0.51\) | M1 | Summing two appropriate 2-factor probabilities |
| A1 | Correct answer | |
| (iii) Variance (number of times) = 7.2 | B1 | Correct final answer |
**(i)** $P(> 5) = ^7C_6(0.6)^6(0.4) + (0.6)^7 = 0.1306 + 0.02799 = 0.159$ | M1 | Summing 2 or 3 binomial probs of the form $^nC_r(0.6)^r(0.4)^{n-r}$ |
| A1 | Correct answer |
**(ii)** $P(\text{bark}) = P(\text{park, bark}) + P(\text{not park, bark}) = 0.6 \times 0.35 + 0.4 \times 0.75 = 0.51$ | M1 | Summing two appropriate 2-factor probabilities |
| A1 | Correct answer |
**(iii)** Variance (number of times) = 7.2 | B1 | Correct final answer |
---3 Christa takes her dog for a walk every day. The probability that they go to the park on any day is 0.6 . If they go to the park there is a probability of 0.35 that the dog will bark. If they do not go to the park there is a probability of 0.75 that the dog will bark.\\
(i) Find the probability that they go to the park on more than 5 of the next 7 days.\\
(ii) Find the probability that the dog barks on any particular day.\\
(iii) Find the variance of the number of times they go to the park in 30 days.
\hfill \mbox{\textit{CAIE S1 2010 Q3 [5]}}