Decision Making with Probabilities (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note
Exam code: 1350
Decision Making with Probabilities
How can I use expected values to help make decisions?
Expected values and probabilities are essential when assessing potential risks and benefits
For example:
Purchasing stocks & shares
Should you invest in a company with potentially large returns, but a high risk of failure?
or a company with a low risk of failure, but smaller returns?
Selecting a mortgage product
Should you select a 2-year fixed mortgage rate of 5% ?
or a 5-year fixed mortgage rate of 4.5% ?
This will depend on the expected values of mortgage rates in 2 and 5 years time
Selecting an energy price tariff
Should you fix your energy price for 12 months?
or should you stick with the 'variable rate' ?
This will depend on the expected value for energy prices over the next 12 months
Insurance on a newly purchased item
When purchasing a new appliance like a washing machine, you may be offered insurance
The probability of the appliance breaking, and its cost to replace, could be used to find the expected cost over say 5 years
This can then be compared to the price for insurance to make a decision
Probabilities can be calculated or estimated to find expected values
The expected values can then be used to help make a decision
E.g. The expected share price after 12 months can help decide which company to invest in
Even when expected values are calculated, there will always be a subjective decision made
This depends on the individual or organisation's appetite for risk
E.g. A low-risk company may have a higher expected share price after 12 months than a high-risk company
But if the high-risk company succeeds, the benefits could be much more lucrative
Worked Example
A football team, SME United, is scheduling two exhibition matches with the aim of raising as much money as possible.
Tickets cost £15 per person.
Previous data suggests that when the SME United win, supporters spend, on average, £8 per person on food, drinks and merchandise.
However when SME United does not win, supporters only spend, on average, £4 per person.
Use the information above, and the predicted information in the table below, to find the predicted income from playing each team.
Hence recommend which two teams SME United should play.
Opposing Team | Probability of SME United winning | Predicted Attendance |
|---|---|---|
Manchester Super Reds | 0.1 | 4000 |
Dalston Hotspur | 0.3 | 3000 |
Aston Village Pub | 0.8 | 3500 |
Answer:
First consider Manchester Super Reds
Find the income from tickets
4000 × £15 = £60 000
Calculate the expected average spend per person using combined probability
Multiply the probability of a win, by the expected income for a win, and add it to the same calculation, but for not winning
The probability of not winning, will be (1 - the probability of winning)
(0.1 × £8) + (0.9 × £4) = £4.40
expected spend per person
Multiply this by the attendance
£4.40 × 4000 = £17 600
Find the total earnings
£60 000 + £17 600
£77 600 earned from playing Manchester Super Reds
Perform the same calculation for Dalston Hotspur
3000 × £15 = £45 000 ticket sales
(0.3 × £8) + (0.7 × £4) = £5.20
expected spend per person
£5.20 × 3000 = £15 600
Total: £45 000 + £15 600
£60 600 earned from playing Dalston Hotspur
Perform the same calculation for Aston Village Pub
3500 × £15 = £52 500 ticket sales
(0.8 × £8) + (0.2 × £4) = £7.20
expected spend per person
£7.20 × 3500 = £25 200
Total: £52 500 + £25 200
£77 700 earned from playing Aston Village Pub
Write a short conclusion to answer the question
I would recommend that SME United schedules their two matches against Manchester Super Reds and Aston Village Pub, in order to make the most money.
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