Probability question

I have jar with infinite number of marbles. 1% of the marbles are red, the remaining 99% are black.

Case 1: If I randomly withdraw 1000 marbles from the jar, I will get approximately 100 red marbles. Right?

Case 2: I withdraw 5 ( = N) marbles at a time and repeat that 10000 times. Each time I am looking for at least one red marble in the 5 that withdrew. So, on average, how may red marbles will I get in the 10000 trials? (In other words, what is the probability of getting at least one marble when N = 5?)

Case 3: Same as case 2, except I make N a random variable, Poisson distributed with a mean value of 5?

Provide mathematical solution for each of the three cases. Provide answer in terms of N. Provide references for the equations.

Case 1: I think the answer is simply 10 or 100*0.01.

Case 2:

Case 3:

Taidot: Algoritmi, Matematiikka, Matlab ja Mathematica, Tilastoanalyysi, Tilastotiede

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