3 Magnus has been researching career possibilities. He has just completed his GCSEs, and could leave school and get a good job. He estimates, discounted at today's values and given a 49 year working life, that there is a $50 \%$ chance of such a job giving him lifetime earnings of $\pounds 1.5 \mathrm {~m}$, a $30 \%$ chance of $\pounds 1.75 \mathrm {~m}$, and a $20 \%$ chance of $\pounds 2 \mathrm {~m}$.

Alternatively Magnus can stay on at school and take A levels. He estimates that, if he does so, there is a 75\% chance that he will achieve good results. If he does not achieve good results then he will still be able to take the same job as earlier, but he will have lost two years of his lifetime earnings. This will give a $50 \%$ chance of lifetime earnings of $\pounds 1.42 \mathrm {~m}$, a $30 \%$ chance of $\pounds 1.67 \mathrm {~m}$ and a $20 \%$ chance of $\pounds 1.92 \mathrm {~m}$.

If Magnus achieves good A level results then he could take a better job, which should give him discounted lifetime earnings of $\pounds 1.6 \mathrm {~m}$ with $50 \%$ probability or $\pounds 2 \mathrm {~m}$ with $50 \%$ probability. Alternatively he could go to university. This would cost Magnus another 3 years of lifetime earnings and would not guarantee him a well-paid career, since graduates sometimes choose to follow less well-paid vocations. His research shows him that graduates can expect discounted lifetime earnings of $\pounds 1 \mathrm {~m}$ with $20 \%$ probability, $\pounds 1.5 \mathrm {~m}$ with $30 \%$ probability, $\pounds 2 \mathrm {~m}$ with $30 \%$ probability, and $\pounds 3 \mathrm {~m}$ with $20 \%$ probability.\\
(i) Draw up a decision tree showing Magnus's options.\\
(ii) Using the EMV criterion, find Magnus's best course of action, and give its value.

Magnus has read that money isn't everything, and that one way to reflect this is to use a utility function and then compare expected utilities. He decides to investigate the outcome of using a function in which utility is defined to be the square root of value.\\
(iii) Using the expected utility criterion, find Magnus's best course of action, and give its utility.\\
(iv) The possibility of high earnings ( $\pounds 3 \mathrm {~m}$ ) swings Magnus's decision towards a university education. Find what value instead of $\pounds 3 \mathrm {~m}$ would make him indifferent to choosing a university education under the EMV criterion. (Do not change the probabilities.)

\hfill \mbox{\textit{OCR MEI D2 2011 Q3 [20]}}