$8.$ Imagine we are given a choice between the following two investment options:

Investment A: With $100\backslash \%$ certainty, we receive a payout of $\backslash (\backslash$$ $1\backslash )$ million.

Investment B: With a $\backslash (10\backslash \%\backslash )$ chance, we receive $\backslash (\backslash$$ $5\backslash )$ million. With an $\backslash (89\backslash \%\backslash )$ chance, we receive $\backslash (\backslash$$ $1\backslash )$ million. With a $\backslash (1\backslash \%\backslash )$ chance, we receive nothing.

Which investment would we prefer?

Now onsider two other investment opportunities:

Investment C: With an $\backslash (11\backslash \%\backslash )$ chance, we receive $\backslash (\backslash$$ $1\backslash )$ million, and with an $\backslash (89\backslash \%\backslash )$ chance, we receive nothing.

Investment D: With a $\backslash (10\backslash \%\backslash )$ chance, we receive $\backslash (\backslash$$ $5\backslash )$ million, and with a $\backslash (90\backslash \%\backslash )$ chance, we receive nothing.

Which of these would we prefer?

Suppose that, like most people, we prefer Investment A over Investment B$.$ Show that this preference implies Investment C must have a larger expected utility than Investment D $.$