A gambler (call him Bob) is playing at a casino thefollowing game: Bob can bet a

Question

A gambler (call him Bob) is playing at a casino thefollowing game: Bob can bet any amount he wishes, then a fair coin istossed, and if the outcome of the toss is “heads” he wins an amount equal tohis bet, otherwise he loses the amount he bet. Bob can play as many rounds of this game as he wishes.Bob decides to follow the following strategy: He bets $100 in round 1, and if he losesround 1 then he doubles the bet for the next round, and he keeps doubling the bet from oneround to the next until he wins; he stops as soon as he wins a round, collects his money,and walks out of the casino. 1. What is the probability that Bob loses all of the first krounds? 2. If Bob has lost the first k rounds and won round k + 1,what is his net gain out of playing these k + 1 rounds?
can i use geometry sequence for this? A gambler (call him Bob) is playing at a casino thefollowing game: Bob can bet any amount he wishes, then a fair coin istossed, and if the outcome of the toss is “heads” he wins an amount equal tohis bet, otherwise he loses the amount he bet. Bob can play as many rounds of this game as he wishes.Bob decides to follow the following strategy: He bets $100 in round 1, and if he losesround 1 then he doubles the bet for the next round, and he keeps doubling the bet from oneround to the next until he wins; he stops as soon as he wins a round, collects his money,and walks out of the casino. 1. What is the probability that Bob loses all of the first krounds? 2. If Bob has lost the first k rounds and won round k + 1,what is his net gain out of playing these k + 1 rounds?
can i use geometry sequence for this?

Explanation / Answer

1) Probability that Bob loses all of first k rounds = P(he losesfirst round)*P(he loses second round)*....P(he loses kth round)

andeach round is independent of the other and has probability 1/2.

therefore = (1/2)k .

2) ifBob has lost K rounds he would lose 100 + 100*2+100*22+....100 * 2(k-1)

= 100( 1+2+22+.... 2(k-1) ) the term inthe braces is a sum of geometric progression with a =1 and r =2 andn = k and k >0

= 100* 1* (2k -1)/(2-1) = 100*(2k -1)

andhe gain 2* {100 (2k )}this is because he getsdouble his bet amount if he wins. (be careful while writing generalterms and check by putting in some sample numbers for k . for eg:if he had won 2nd bet he should have got 100*2 = $ 200

therefore total gain = -100 *(2k -1)+100(2k+1 ) k>0

eg:if he loses 1st one and wins 2nd one…he loses 100 and gains 200*2 =400. So total gain = 300 . Andformula that we have derived give -100(21 -1)+100(22 ) = -100+400 =300.

hopethis helps. Feel free to ask for any clarifications.

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