An urn has 7 red marbles and 13 blue marbles. Careful on this one, the distribution to use keeps changing!

• If I pick 4 marbles with replacement and let X= the number of red marbles picked, what is P(X=1)?
• If I keep picking marbles with replacement until I get my first red marble, what is the probability I picked exactly 7 times?
• If I pick 4 marbles without replacement and let X= the number of red marbles picked, what is the P(X=2)?
• If I pick 4 marbles without replacement and let X= the number of red marbles picked, what is the expected value of X?
• Suppose I have picked 20 marbles with replacement and have not yet gotten a red marble. How many more picks do I expect will be required before my first red marble?
• Suppose I have picked 10 marbles without replacement and have not yet gotten a red marble. How many more picks (without replacement) do I expect will be required before my first red marble? This is a tricky problem! We can't use any of our favorite distributions; you have to think through it from first principles. Here's a hint to get you started. Let X = number of remaining picks. Compute P(X=1), P(X=2), P(X=3), etc. (it will help to think of the Geometric distribution, but it's not quite the same) and then compute the mean value from there.