OCR MEI S1 (Statistics 1)

1 A survey is being carried out into the sports viewing habits of people in a particular area. As part of the survey, 250 people are asked which of the following sports they have watched on television in the past month.

• Football
• Cycling
• Rugby

The numbers of people who have watched these sports are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{870b6ef1-60f7-42e3-95f8-0544a2a07b15-1_725_921_728_622} One of the people is selected at random.

1. Find the probability that this person has in the past month
   (A) watched cycling but not football,
   (B) watched either one or two of the three sports.
2. Given that this person has watched cycling, find the probability that this person has not watched football.

2 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.

1. Find the probability that neither card is an ace.
2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.

3 Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities \(0.2,0.5\) and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.

1. Draw a probability tree diagram to illustrate the outcomes.
2. Find the probability that a randomly selected candidate is accepted.
3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted.

4 Each weekday, Marta travels to school by bus. Sometimes she arrives late.

• \(L\) is the event that Marta arrives late.
• \(R\) is the event that it is raining.

You are given that \(\mathrm { P } ( L ) = 0.15 , \mathrm { P } ( R ) = 0.22\) and \(\mathrm { P } ( L \mid R ) = 0.45\).

1. Use this information to show that the events \(L\) and \(R\) are not independent.
2. Find \(\mathrm { P } ( L \cap R )\).
3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.

5 Each weekday Alan drives to work. On his journey, he goes over a level crossing. Sometimes he has to wait at the level crossing for a train to pass.

• \(W\) is the event that Alan has to wait at the level crossing.
• \(L\) is the event that Alan is late for work.

You are given that \(\mathrm { P } ( L \mid W ) = 0.4 , \mathrm { P } ( W ) = 0.07\) and \(\mathrm { P } ( L \cup W ) = 0.08\).

1. Calculate \(\mathrm { P } ( L \cap W )\).
2. Draw a Venn diagram, showing the events \(L\) and \(W\). Fill in the probability corresponding to each of the four regions of your diagram.
3. Determine whether the events \(L\) and \(W\) are independent, explaining your method clearly.

6 Malik is playing a game in which he has to throw a 6 on a fair six-sided die to start the game. Find the probability that

1. Malik throws a 6 for the first time on his third attempt,
2. Malik needs at most ten attempts to throw a 6.

7 At a garden centre there is a box containing 50 hyacinth bulbs. Of these, 30 will produce a blue flower and the remaining 20 will produce a red flower. Unfortunately they have become mixed together so that it is not known which of the bulbs will produce a blue flower and which will produce a red flower. Karen buys 3 of these bulbs.

1. Find the probability that all 3 of these bulbs will produce blue flowers.
2. Find the probability that Karen will have at least one flower of each colour from her 3 bulbs.

8 At a call centre, 85\% of callers are put on hold before being connected to an operator. A random sample of 30 callers is selected.

1. Find the probability that exactly 29 of these callers are put on hold.
2. Find the probability that at least 29 of these callers are put on hold.
3. If 10 random samples, each of 30 callers, are selected, find the expected number of samples in which at least 29 callers are put on hold.