7. Consider the following game: There are 2 boxes. Each box contains several envelopes.

Some envelopes have "you lose" written on them. The rest say "you win ...". However,

of the ones that say "you win ...", some contain only a piece of paper saying "... nothing",

while each of the others contains a $5 bill.

The first box contains 3 "you lose" envelopes and 2 "you win ..." envelopes. The second

contains 1 "you lose" envelope and 3 "you win ..." envelopes. Each box contains only one $5

bill (in an envelope marked "you win ...").

Playing this game consists of choosing one of the 2 boxes at random, and then drawing one

envelope at random from the chosen box. If the drawn envelope contains a $5 bill, the player

gets to keep it, i.e. wins $5.

(a) Draw a probability tree to model one play of this game.

(b) Find the probability that when you play this game you will win $5.

(c) Your friend Clyde played the game (only once) and got an envelope which said "you win

..." but contained no money. What is the probability that Clyde drew an envelope from

the second box?

(d) You are going to play this game 5 times. After each play, the drawn envelope will be put

back in the box it was drawn from, having been replenished with money if necessary. Do

not simplify your answers to the following.

Note: Do not draw a new tree.

(i) What is the probability that you will win exactly $15?

(ii) What is the probability that you will win exactly $10 and will draw only 1 envelope

which says "you win ..." but contains no money?