Problem about Poker and the probability of certain events

Question

I'm having trouble with this exercise

Consider drawing cards one by one without replacement from a deck of cards. Let Ri denote the event that the ith draw results in a red card.

1. Find P(R2).

2. Find P(R52). How do you compute this one?

3. Find P(R52|R1).

4. Comparing P(R1|R2) with P(R1), which one is larger? Why?

I know how to solve part 1 and 4.

For Part 2 I believe that the answer is 1/2, and for Part 3 should be 25/51. But I don't know how I can prove this. How do you compute this answers? I couldn't find a combination that would give me this answers.

Answer 1

For each part you can ignore all the cards you don't look at. For 2, you can imagine dealing all the cards, then reversing the order. The chance of the first card being red is $\frac 12$, so the chance of the last card being red is also $\frac 12$. For 3, you pull one red card from the deck, then pick a card. The chance it is red is $\frac{25}{51}$. The same reversal argument works here after you pull the first red card. 4 is just like 3. If you have already pulled a red card, the chance of another is decreased, so $P(R1)$ is larger. In fact you are comparing $\frac 12$ and $\frac {25}{51}$, but you weren't asked for that detail.