Why do we use weights OR probabilities in the Expected Utility formula?

[Gerry]

I am confused , we are given that expected utility is:

https://imgur.com/Y7p7jgf

i.e. the probability weighted average of the Utilities with each outcome. This is what I have used for most questions regarding expected utility.

So how come for portfolio related questions we use weights when we are maximising expected utility? I just don't understand why.

For eg. Chapter 2 Q2.1 part (a),

The solution shows we are solving for Xa, the proportion in asset A...

 

[Alvin Kissoon]

Hi Gerry,

You are right that for calculating the expected utility you think of the probability weighted average of the utilities. In these questions, you are given a series of outcomes, each with a certain wealth and an associated probability. You don't need to be given weights because you are given the outcomes in terms of wealth.

The difference with portfolios is that we are often provided utility either where the wealth is expressed in terms of the weights, or in the case of 2.1(a), we are provided utility in terms of expected value and variance. To calculate these, we form formulas for the utility in terms of the weights Xa. Then, to maximise the utility we will maximise this function.

Alvin.