Dice/random device selects population

3 Tim has two bags of marbles, \(A\) and \(B\).
Bag \(A\) contains 8 white, 4 red and 3 yellow marbles.
Bag \(B\) contains 6 white, 7 red and 2 yellow marbles.
Tim also has an ordinary fair 6 -sided dice. He rolls the dice. If he obtains a 1 or 2 , he chooses two marbles at random from bag \(A\), without replacement. If he obtains a \(3,4,5\) or 6 , he chooses two marbles at random from bag \(B\), without replacement.

1. Find the probability that both marbles are white.
2. Find the probability that the two marbles come from bag \(B\) given that one is white and one is red. [4]

5 Playground equipment consists of swings ( \(S\) ), roundabouts ( \(R\) ), climbing frames ( \(C\) ) and play-houses \(( P )\). The numbers of pieces of equipment in each of 3 playgrounds are as follows. Each day Nur takes her child to one of the playgrounds. The probability that she chooses playground \(X\) is \(\frac { 1 } { 4 }\). The probability that she chooses playground \(Y\) is \(\frac { 1 } { 4 }\). The probability that she chooses playground \(Z\) is \(\frac { 1 } { 2 }\). When she arrives at the playground, she chooses one piece of equipment at random.

1. Find the probability that Nur chooses a play-house.
2. Given that Nur chooses a climbing frame, find the probability that she chose playground \(Y\).

2 The people living in two towns, Mumbok and Bagville, are classified by age. The numbers in thousands living in each town are shown in the table below. One of the towns is chosen. The probability of choosing Mumbok is 0.6 and the probability of choosing Bagville is 0.4 . Then a person is chosen at random from that town. Given that the person chosen is between 18 and 60 years old, find the probability that the town chosen was Mumbok. [5]

6\% of the counters labelled C are red
One counter is selected at random from the bag.

1. Complete the tree diagram on the opposite page to illustrate this information.
2. Calculate the probability that the counter is labelled A and is not red.
3. Calculate the probability that the counter is red.
4. Given that the counter is red, find the probability that it is labelled C \end{enumerate} \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-15_1155_1000_285_456}
   5. A discrete random variable \(Y\) has probability function $$\mathrm { P } ( \mathrm { Y } = \mathrm { y } ) = \left\{ \begin{array} { c l } \mathrm { k } ( 3 - \mathrm { y } ) & y = 1,2
   \mathrm { k } \left( \mathrm { y } ^ { 2 } - 8 \right) & y = 3,4,5
   \mathrm { k } & y = 6
   0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.
5. Show that \(k = \frac { 1 } { 30 }\) Find the exact value of
6. \(\mathrm { P } ( 1 < Y \leqslant 4 )\)
7. \(\mathrm { E } ( Y )\) The random variable \(X = 15 - 2 Y\)
8. Calculate \(\mathrm { P } ( Y \geqslant X )\)
9. Calculate \(\operatorname { Var } ( X )\)
   1. Three events \(A , B\) and \(C\) are such that
   $$\mathrm { P } ( A ) = 0.1 \quad \mathrm { P } ( B \mid A ) = 0.3 \quad \mathrm { P } ( A \cup B ) = 0.25 \quad \mathrm { P } ( C ) = 0.5$$ Given that \(A\) and \(C\) are mutually exclusive
10. find \(\mathrm { P } ( A \cup C )\)
11. Show that \(\mathrm { P } ( B ) = 0.18\) Given also that \(B\) and \(C\) are independent,
12. draw a Venn diagram to represent the events \(A , B\) and \(C\) and the probabilities associated with each region.

2 A hotel chain has hotels in three types of location: city, coastal and country. The percentages of the chain's reservations for each of these locations are 30,55 and 15 respectively. Each of the chain's hotels offers three types of reservation: Bed \& Breakfast, Half Board and Full Board. The percentages of these types of reservation for each of the three types of location are shown in the table. For example, 80 per cent of reservations for hotels in city locations are for Bed \& Breakfast.

1. For a reservation selected at random:
   1. show that the probability that it is for Bed \& Breakfast is 0.34 ;
   2. calculate the probability that it is for Half Board in a hotel in a coastal location;
   3. calculate the probability that it is for a hotel in a coastal location, given that it is for Half Board.
2. A random sample of 3 reservations for Half Board is selected. Calculate the probability that these 3 reservations are for hotels in different types of location.

3 An IT help desk has three telephone stations: Alpha, Beta and Gamma. Each of these stations deals only with telephone enquiries. The probability that an enquiry is received at Alpha is 0.60 .
The probability that an enquiry is received at Beta is 0.25 .
The probability that an enquiry is received at Gamma is 0.15 . Each enquiry is resolved at the station that receives the enquiry. The percentages of enquiries resolved within various times at each station are shown in the table. For example:
80 per cent of enquiries received at Alpha are resolved within 24 hours;
25 per cent of enquiries received at Alpha take between 1 hour and 24 hours to resolve.

1. Find the probability that an enquiry, selected at random, is:
   1. resolved at Gamma;
   2. resolved at Alpha within 1 hour;
   3. resolved within 24 hours;
   4. received at Beta, given that it is resolved within 24 hours.
2. A random sample of 3 enquiries was selected. Given that all 3 enquiries were resolved within 24 hours, calculate the probability that they were all received at:
   1. Beta;
   2. the same station.
      \begin{center} \begin{tabular}{|l|l|l|} \hline & & \begin{tabular}{l}

3 A hotel has three types of room: double, twin and suite. The percentage of rooms in the hotel of each type is 40,45 and 15 respectively. Each room in the hotel may be occupied by \(0,1,2\), or 3 or more people. The proportional occupancy of each type of room is shown in the table.

3 A particular brand of spread is produced in three varieties: standard, light and very light. During a marketing campaign, the producer advertises that some cartons of spread contain coupons worth \(\pounds 1 , \pounds 2\) or \(\pounds 4\). For each variety of spread, the proportion of cartons containing coupons of each value is shown in the table. For example, the probability that a carton of standard spread contains a coupon worth \(\pounds 2\) is 0.08 . In a large batch of cartons, 55 per cent contain standard spread, 30 per cent contain light spread and 15 per cent contain very light spread.

1. A carton of spread is selected at random from the batch. Find the probability that the carton:
   1. contains standard spread and a coupon worth \(\pounds 1\);
   2. does not contain a coupon;
   3. contains light spread, given that it does not contain a coupon;
   4. contains very light spread, given that it contains a coupon.
2. A random sample of 3 cartons is selected from the batch. Given that all of these 3 cartons contain a coupon, find the probability that they each contain a different variety of spread.
   [0pt] [4 marks]

3 Each enquiry received by a business support unit is dealt with by Ewan, Fay or Gaby. The probabilities of them dealing with an enquiry are \(0.2,0.3\) and 0.5 respectively. Of enquiries dealt with by Ewan, 60\% are answered immediately, 25\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Fay, 75\% are answered immediately, 15\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Gaby, 90\% are answered immediately and the remainder are answered at a later date.

1. Determine the probability that an enquiry:
   1. is dealt with by Gaby and answered immediately;
   2. is answered immediately;
   3. is dealt with by Gaby, given that it is answered immediately.
2. Determine the probability that an enquiry is dealt with by Ewan, given that it is answered later the same day.

2 A hill-top monument can be visited by one of three routes: road, funicular railway or cable car. The percentages of visitors using these routes are 25, 35 and 40 respectively. The age distribution, in percentages, of visitors using each route is shown in the table. For example, 15 per cent of visitors using the road were under 18 . Calculate the probability that a randomly selected visitor:

1. who used the road is aged 18 or over;
2. is aged between 18 and 64;
3. used the funicular railway and is aged over 64;
4. used the funicular railway, given that the visitor is aged over 64.