Five problems, or groups of problems, are introduced here. The intention is for the reader to investigate, verify, and add to or extend these problems. In many cases, results are stated without proof, and in these cases proofs may be difficult. Mathematica has been used widely and this tool, or a computer equivalent, will be very useful in achieving these goals.
I have 7 pairs of socks in a drawer. I do not sort them by pairs, so they are randomly distributed in the drawer. I select socks at random until I have a pair. The probability I get a pair in 2 drawings is , since the first sock can be any one of 14 socks and the second sock must be the other sock in the pair represented by the first sock.
The probability it takes 3 draws to get a pair is , since the first sock can be any one of the 14 socks in the drawer, the second sock must not match the first, and the third sock can match either of the first two socks drawn.
In a similar way, we can find the probability distribution of the random variable , the number of draws it takes ...