Decision-making with random term

Question

Consider the following situation.

There are multiple options to choose from based on an attribute related to those options. For example:

option   attribute
A        30
B        20

Whichever has the highest value for the attribute gets chosen. But a random term is introduced from the normal distribution and added to the attribute.

If for example we use N(0, 7.75), we get a probability of around 10% that the random term is less than -10. So there's a probability of around 10% that the attribute value for A is less than 20.

But we also add the random term for B, so there's a probability of around 50% that the random term is positive and thus the attribute value for B is 20 or more.

In this situation we would get that the probability for A to get chosen rationally based on the original attribute values is 10% * 50% = 5%.

Taking into account that the same can also happen the other way, so that B gets a random term of 10 or more and A gets a random term that is 0 or negative, we also have a 5% probability for B to get chosen irrationally like this.

My question is:

Would I be correct to say that the decision-making here is rational 90% of the time?

Answer 1

You're not doing this well.

I think that A~N(30,7.75) and B~N(20,7.75). Let C = A - B. Find the distribution of C. Irrational choice is P(C<0).