Problem $3\mathrm{.}9($Change making problem$)$: Dani Hofmann, who is working at a local retail store, is always busy making changes with coins daily. Naturally, he is interested in the average number of coins required that can make any change from $0$ to $99$ cents.

Suppose that there are $1$ cent, $5$ cent, $10$ cent, $20$ cent, $25$ cent, and $50$ cent pieces. Let us assume that the probability of a transaction resulting in value $x$ has a uniform distribution over $0,99.($You may use any spreadsheet software or write your own script to answer the following questions.$)(4$ points$)$

$($a$)$ If he always picks the largest denomination of coin first, find the average number of coins in a transaction.

$($b$)$ If he uses the most efficient dynamic programming method, what is the average number of coins in a transaction?