1. Expected value

Assume there are 700 colors. 322 reds. 336 blues. 42 neutrals. We sample 70 of them.

a) What is the probability of only getting only blues and reds?
b) What is the expected value of the number of simple random samples of size 70 that must be drawn until (and including) the first sample that contains at least one color that is neither a blue nor a red if you repeatedly drawing simple random samples of size 70 (taking each simple random sample from the full population of 700)?

a) .01038
b) 1.01048887451749

Can somebody explain to me how it's done?

2. Hello, Starry!

Where did this problem come from?

Part (a) has an answer that no one wants to crank out.

There are 700 marbles: 322 reds. 336 blues. 42 neutrals. We sample 70 of them.

a) What is the probability of only getting only blues and reds?

There are 700 marbles; we sample 70 of them.
. . There are: .$\displaystyle {700\choose70}$ possible samples.

There are 658 marbles that are red or blue; we want 70 of them.
. . There are: .$\displaystyle {658\choose70}$ samples that have only blues and/or reds.

The probability is: .$\displaystyle \frac{{658\choose70}} {{700\choose70}} \;=\;\frac{658!}{70!\,588!}\,\frac{70!\,630!}{700! } \;=\;\frac{630!}{588!}\cdot \frac{658!}{700!}$

. . And the answer is: .$\displaystyle \frac{630\cdot629\cdot629\cdots589}{700\cdot699\cd ot698\cdots 659}$

I'll wait in the car . . .
.