A gambler plays two games of chance. Every time she plays Game A, she has a chance1/5n of winning, regardless of the outcomes of all other games. Every time she plays Game B, she has a chance1/n of winning, regardless of the results of all other games. She has decided on the following strategy.

• She keeps playing Game A until either she wins or she has played Game An times and lost every time.
• If she loses all n times on Game A, she starts playing Game B. She keeps playing until either she wins or she has played Game B n times and lost every time.

a)What is the chance that the gambler doesn't have to play Game B?

b)LetW be the event that the gambler ends on a win. PartitionW by filling in the blank with a verbal description of an event, and then use the partition to findP(W).

W={doesn't have to play Game B} or {}

c)The complementWc can be written asWc=AnBn whereAn is an event based only on playing Game A n times andBn is an event based only on playing Game B n times. Give verbal descriptions ofAnandBn, and hence useWc=AnBn to findP(W). Show that the answer is equal to the answer in Partb.

d)Assumen is large, and find an exponential approximation to the chance that the gambler ends on a win. After you have shown the math, in the cell below write an expression the evaluates to the approximate numerical value of the chance.