Week 1 Probability Tutorial PDF

Quantitative Methods Week 1 Tutorial: Probability 1. You are told that you need surgery on your knee and will spend between 1 and 3 days in hospital. Hospital records show the number of patients required to stay 1, 2, or 3 days for the same operation are: D...

Quantitative Methods Week 1 Tutorial: Probability 1. You are told that you need surgery on your knee and will spend between 1 and 3 days in hospital. Hospital records show the number of patients required to stay 1, 2, or 3 days for the same operation are: Days in hospital Number of patients 1 700 2 350 3 150 What is the probability you will need 3 days in hospital? 2. The career destinations of an Accountancy graduate from a university are: Destination Probability Accountancy 0.5 Insurance 0.05 Banking 0.05 Other finance 0.1 University teaching 0.05 Teaching 0.1 Other 0.15 1 (a) What is the probability a graduate goes into any sort of teaching? (b) What is the probability a graduate does not go into accountancy? 3. Toss three coins. 1st coin 2nd coin Outcome 1 Head Head Outcome 2 Head Tail Outcome 3 Tail Head Outcome 4 Tail Tail List all possible outcomes and their probabilities. Calculate the probability of obtaining exactly (i) one heads, (iii) two heads, (iii) three heads. 4. There are 1,500 students on a university degree course: Accountancy Economics Finance BIS Male 330 360 90 120 Female 120 390 60 30 Using this data, if you selected a student at random … (a) What is the probability that the student is doing an economics degree? (b) What is the probability that the student is male? (c) What is the probability that the student is female and doing economics or finance? 5. What is the probability of getting a heart when selecting a card from a pack of 52 cards? 6. In the last 100 working days the stock market has risen for 6 days, remained the same for 20 days and dropped for 74 days. At the close of business today it is 2721.  Assuming conditions remain the same, what is the probability that it will be greater than 2721 at the close of business tomorrow? 7. A bank made 500 car loans last year. The amounts were as follows: £ No. of Loans Under 1,000 27 1,000 – 3,999 99 4,000 – 5,999 298 6,000+ 76 One loan is sampled at random. What is the probability it is: a) Under £1,000? b) Greater than or equal to £4,000? 8. A couple plan to have two children. Assume a boy and a girl is equally likely at each birth: (a) List the possible outcomes (b) What is the probability for these outcomes? (c) What is the probability they will have two boys? 9. A computer retailer conducts a survey of the ages of 200 computer purchasers: Less than 30 30 – 44 45 and over Male 60 20 40 Female 40 30 10 A customer is selected at random. 10. If the selected customer is aged 30 – 44 what is the probability that they are male? A business produces CDs at two factories. 40% of all CDs are produced at factory A and the rest at factory B. About 2% of CDs produced at factory A are defective. What is the probability that a CD is defective given that it was manufactured at factory A? 11. A fast-food chain has grouped their outlets into four geographical areas: Population: Regions: NE SE SW NW Under 10,000 35 42 21 70 10,000 – 100,000 70 105 84 35 Over 100,000 175 28 35 0 If an outlet were to be selected for an inspection at random: a) What is the probability that an outlet in the NE is chosen? b) It is known that the outlet to be inspected is in an area with a population over 100,000. What is the probability that the outlet is in the NE? c) Are the events ‘outlet in the NE’ and ‘in an area with a population over 100,000’ independent events? 12. Forty percent of the UK population are under 20. Of these 60% watch ‘Neighbours’, a television program. If a person from the UK is selected at random, what is the probability that he or she is under 20 and watches ‘Neighbours’? 13. A student wants to see 6 out of 20 films being shown by the student union on a Friday and Saturday night this term. If the student does not go to a film, there is a 30% chance he will go to the union bar. If he goes to the film, there is a 60% chance he will go to the union bar afterwards. What is the probability that on a Friday or Saturday night this term, the student will go a film and then to the bar? 14. When the weather is stormy the probability that the lifeboat service is called out is 40%. When the weather is not stormy the probability is 10%. It is stormy about 1 night in 20. What is the probability that it is stormy and the lifeboat is called out on a particular night Probability formulae Addition rule: 𝑷(𝑨 ∪ 𝑩) = 𝑷(𝑨) + 𝑷(𝑩) − 𝑷(𝑨 ∩ 𝑩) Multiplication rule: 𝑷(𝑨 ∩ 𝑩) = 𝑷(𝑨) 𝑷(𝑩 |𝑨) Or 𝑷(𝑨 ∩ 𝑩) = 𝑷(𝑩) 𝑷(𝑨|𝑩) Complements: 𝑷(𝑨 ) = 𝟏 − 𝑷(𝑨)