CAIE S1 (Statistics 1) 2017 June

1 A biased die has faces numbered 1 to 6 . The probabilities of the die landing on 1,3 or 5 are each equal to 0.1 . The probabilities of the die landing on 2 or 4 are each equal to 0.2 . The die is thrown twice. Find the probability that the sum of the numbers it lands on is 9 .

2 The probability that George goes swimming on any day is \(\frac { 1 } { 3 }\). Use an approximation to calculate the probability that in 270 days George goes swimming at least 100 times.

3 A shop sells two makes of coffee, Café Premium and Café Standard. Both coffees come in two sizes, large jars and small jars. Of the jars on sale, \(65 \%\) are Café Premium and \(35 \%\) are Café Standard. Of the Café Premium, 40\% of the jars are large and of the Café Standard, 25\% of the jars are large. A jar is chosen at random.

1. Find the probability that the jar is small.
2. Find the probability that the jar is Café Standard given that it is large.

4

1. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(\mu = 1.5 \sigma\). A random value of \(X\) is chosen. Find the probability that this value of \(X\) is greater than 0 .
2. The life of a particular type of torch battery is normally distributed with mean 120 hours and standard deviation \(s\) hours. It is known that \(87.5 \%\) of these batteries last longer than 70 hours. Find the value of \(s\).

5 Hebe attempts a crossword puzzle every day. The number of puzzles she completes in a week (7 days) is denoted by \(X\).

1. State two conditions that are required for \(X\) to have a binomial distribution.
   On average, Hebe completes 7 out of 10 of these puzzles.
2. Use a binomial distribution to find the probability that Hebe completes at least 5 puzzles in a week.
3. Use a binomial distribution to find the probability that, over the next 10 weeks, Hebe completes 4 or fewer puzzles in exactly 3 of the 10 weeks.

6

1. Find how many numbers between 3000 and 5000 can be formed from the digits \(1,2,3,4\) and 5,
   1. if digits are not repeated,
   2. if digits can be repeated and the number formed is odd.
2. A box of 20 biscuits contains 4 different chocolate biscuits, 2 different oatmeal biscuits and 14 different ginger biscuits. 6 biscuits are selected from the box at random.
   1. Find the number of different selections that include the 2 oatmeal biscuits.
   2. Find the probability that fewer than 3 chocolate biscuits are selected.

7 The following histogram represents the lengths of worms in a garden.
\includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-10_789_1195_301_466}

1. Calculate the frequencies represented by each of the four histogram columns.
2. On the grid on the next page, draw a cumulative frequency graph to represent the lengths of worms in the garden.
   \includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-11_1111_1409_251_408}
3. Use your graph to estimate the median and interquartile range of the lengths of worms in the garden.
4. Calculate an estimate of the mean length of worms in the garden.
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