Card game questions

Question

I write down 4 numbers on four cards and you should choose one card. After seeing the card you can decide to throw away this card and pick another one without replacement. You can stop anytime you want, but if your card has the highest number among four you will win 6$. Calculate the expected value. Say what is the best strategy. I solved this problem for n=2 and the expected value is 1.8. However, I could not generalize the answer for n=100. Could you please help me to do that.

Answer 1

Mihri, what do you think about this calculation of the expected value:

$$\left(1\times \frac{1}{4}\right)+\left(2\times \frac{1}{4}\right)+\left(3\times \frac{1}{4}\right)+\left(4\times \frac{1}{4}\right)=\frac{10}{4}$$