Problem $2$: Conditional probabilities, total probability theorem, efficient markets?

The dudes at NFQ Ltd$.($No Finance Quacks$)$ know that investors care whether the market

return is positive. It$$s probably all they know. They want to sell a weekly financial letter that

predicts the sign of the weekly US Market return. They consider $3$ strategies:

$1)$ Always predict $>0$ return, $2)$ Always predict $<0$ return,

$3)$ Randomly predicts $>0$ return $60\%$ of the time. Dudes read somewhere that this is called the

$$randomization strategy$$ and relates to $$adaptive expectations$,$ big thing in psychology $!$

They read your published research showing that the weekly market excess return is positive

with probability p$($U$)=59\%.$ They found this interesting, so they hired you as a consultant for

this project.

a$)$ Give them the weekly probability of correct prediction and the expected number of correct

predictions after $52$ weeks for each rule. Denote $$C$$ the event $$correct prediction$,$U$$:

market$$s up$,$D$$: market$$s down. Round up all probabilities to $2$ decimals maximum.

Write the formula you will use: p$($C$)=$

Rule $1$: p$($C$)=$

Rule $2$: p$($C$)=$

Rule $3$: p$($C$)=$

b$)$ Explain in words, what is wrong with the $$randomizing strategy$(3).$

$4$