A child has to do the following experiment as part of their math homework: "Toss a fair coin until you get a head, and then stop tossing. Report the number of times you tossed."

The child is grumbling and refusing to do. So their parent decides to bribe them.

a) The parent tells the child, "I'll give you one M&M for each toss! For example, if you toss 5 times to get a head, you'll get 5 M&Ms!" The trouble is, the parent only has 7 M&M's, so if the number of tosses until a head is more than 7 then the child will get only 7 M&Ms.

What is the expected number of M&Ms the child will get if they accept the parent's offer? Write math formula for the expectation with no infinite sums in your answer. Then provide its numerical value, using the cell below for your calculation if necessary.

b) The child is not happy that they can get at most 7 M&Ms, and rejects the offer in Part a. So the parent buys a big bag of M&Ms and tells the child, "However many tosses it takes you to get a head, I'll give you 2 times that many M&Ms! For example, if it takes you 55 tosses, you'll get 25=1025=10 M&Ms!"

What is the expected number of M&Ms the child will get if they accept the parent's offer? Write math formula for the expectation with no infinite sums in your answer. Then provide its numerical value, using the cell below for your calculation if necessary.

c) The child realizes that rejecting the parent's offers leads to better offers, so they reject the offer in Part b as well. In desperation, the parent says, "OK, however many tosses it takes you to get a head, I'll give you 2 to the power that many M&Ms! For example, if you it takes you 55 tosses to get a head, you'll get 25=3225=32 M&Ms!"

Try to find the expected number of M&Ms the child will get, and explain why the parent is in big trouble (rightly so, for bribing their kid to do homework).