Because you only asked about parts d) and e), I'll only address those.
d) There are 5 coins in the bag (four fair coins and one biased coin). If you select 4 of those 5 coins, the probability of the biased coin being selected is, naturally, 4/5.
e) First, I'll describe what statisticians mean by "expected value". When we say "expected value" we mean "average value", not "most likely value". As a silly example, suppose the census showed that in a city with 100,000 households, there were a total of 230,000 children. This would mean that the expected number of children in a given household was 2.3. However, if a single household was found to have 2.3 children, this would be surprising, not expected, and would prompt questions as to what the extra 30% of a child looked like and if the remaining 70% of the child were buried in the backyard somewhere.
Returning to our coins-in-the-bag example, here's how to calculate the average number of heads.
Each coin has a 20% chance of being the biased coin, in which case it will flip heads 75% of the time.
Each coin has an 80% chance of being a fair coin, in which case it will flip heads 50% of the time.
Using the law of total probability, the probability of a given coin flipping heads if we do not know whether or not the coin is biased can be calculated with the formula
p = 0.2*0.75 + 0.8*0.5 = 0.55
So, each coin has a 55% chance of flipping heads.
We flip 4 coins, each with a 55% chance of flipping heads. The average total number of heads is then
4*0.55 = 2.20
Final answer: 2.20 average heads.
