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### MIKE

#### Cambridge International AS and A Level

 Name of student MIKE Date Adm. number Year/grade 1983 Stream Mike Subject Probability & Statistics 1 (S1) Variant(s) P61, P62, P63 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 11 11 11 10 9 11 63
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/61/M/J/14/7, Topic: Representation of data A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarised in the table below. $$\begin{array}{|l|c|c|c|c|c|} \hline \text{Number of typing errors} & 1-5 & 6-20 & 21-35 & 36-60 & 61-80 \\ \hline \text{Frequency} & 24 & 9 & 21 & 15 & 42 \\ \hline \end{array}$$$\text{(i)}$Draw a histogram on graph paper to represent this information.$[5]\text{(ii)}$Calculate an estimate of the mean number of typing errors for these 111 people.$[3]\text{(iii)}$State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range.$[3]$Question 2 Code: 9709/62/M/J/14/7, Topic: The normal distribution The time Rafa spends on his homework each day in term-time has a normal distribution with mean$1.9$hours and standard deviation$\sigma$hours. On$80 \%$of these days he spends more than$1.35$hours on his homework.$\text{(i)}$Find the value of$\sigma$.$[3]\text{(ii)}$Find the probability that, on a randomly chosen day in term-time, Rafa spends less than 2 hours on his homework.$[2]\text{(iii)}$A random sample of 200 days in term-time is taken. Use an approximation to find the probability that the number of days on which Rafa spends more than$1.35$hours on his homework is between 163 and 173 inclusive.$[6]$Question 3 Code: 9709/63/M/J/14/7, Topic: Permutations and combinations Nine cards are numbered$1,2,2,3,3,4,6,6,6\text{(i)}$All nine cards are placed in a line, making a 9-digit number. Find how many different 9-digit numbers can be made in this way$\text{(a)}$if the even digits are all together,$[4]\text{(b)}$if the first and last digits are both odd.$[3]\text{(ii)}$Three of the nine cards are chosen and placed in a line, making a 3-digit number. Find how many different numbers can be made in this way$\text{(a)}$if there are no repeated digits,$[2]\text{(b)}$if the number is between 200 and 300.$[2]$Question 4 Code: 9709/61/O/N/14/7, Topic: Permutations and combinations A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done$\text{(i)}$if there are more women than men on the committee,$[4]\text{(ii)}$if the committee consists of 3 men and 3 women but two particular men refuse to be on the committee together?$[3]$One particular committee consists of 5 women and 1 man$\text{(iii)}$In how many different ways can the committee members be arranged in a line if the man is not at either end?$[3]$Question 5 Code: 9709/62/O/N/14/7, Topic: Discrete random variables In Marumbo, three quarters of the adults own a cell phone.$\text{(i)}$A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive.$[3]\text{(ii)}$A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone.$[5]\text{(iii)}$Justify the use of your approximation in part$\text{(ii)}$.$[1]$Question 6 Code: 9709/63/O/N/14/7, Topic: Probability A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable$X$is the number of apples which have been taken when the process stops.$\text{(i)}$Show that$\mathrm{P}(X=3)=\frac{1}{3}$.$[3]\text{(ii)}$Draw up the probability distribution table for$X$.$[3]$Another box contains 2 yellow peppers and 5 orange peppers. Three peppers are taken at random from the box without replacement.$\text{(iii)}$Given that at least 2 of the peppers taken from the box are orange, find the probability that all 3 peppers are orange.$[5]\$

Worked solutions: P1, P3 & P6 (S1)

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