In the original VNM theorem, lotteries are taken over global states of the world, meaning that preferences are expressed over mutually exclusive states of the world. Assuming there’s no other states of the world besides vanilla (V) and chocolate (C), your original lotteries are:
0.5*u(V) + 0.5u(c) = 1.5 + 4 = 5.4
against
1u(v) + 0u(c) = 3
so your preference goes to the first lottery. In the second set of lotteries you have
In the original VNM theorem, lotteries are taken over global states of the world, meaning that preferences are expressed over mutually exclusive states of the world.
Assuming there’s no other states of the world besides vanilla (V) and chocolate (C), your original lotteries are:
0.5*u(V) + 0.5u(c) = 1.5 + 4 = 5.4
against
1u(v) + 0u(c) = 3
so your preference goes to the first lottery. In the second set of lotteries you have
0.5*u(V) + 0.5u(c) = 3.5 + 6 = 9.4
against
1u(v) + 0u(c) = 7
You continue to prefer the first lottery.