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QuestionIn a coin game, you start betting \$1, doubling after each loss. With a bankroll of \$127, what’s your expected profit?

Studdy Solution

STEP 1

Assumptions1. The initial bet is 1..Ifyoulose,youdoubleyourbet.<br/>3.Youcontinuethisstrategyuntilyouwinatoss.<br/>4.Thetotalbankrollavailableis1. . If you lose, you double your bet.<br />3. You continue this strategy until you win a toss.<br />4. The total bankroll available is 127.
5. The probability of winning (landing on Heads) and losing (landing onails) in a coin toss is equal (0.5 each).

STEP 2

First, let's calculate the maximum number of rounds you can play with the given bankroll. This is the highest power of2 that is less than or equal to the bankroll.
Rounds=⌊log⁡2(Bankroll)⌋Rounds = \lfloor \log2(Bankroll) \rfloor

STEP 3

Plug in the given value for the bankroll to calculate the maximum number of rounds.
Rounds=⌊log⁡2(127)⌋Rounds = \lfloor \log2(127) \rfloor

STEP 4

Calculate the maximum number of rounds.
Rounds=7Rounds =7

STEP 5

Now, let's calculate the expected profit. The expected profit is the sum of the profits for each round multiplied by the probability of that profit occurring.
Expected profit=∑i=0RoundsProfiti×ProbabilityiExpected\, profit = \sum_{i=0}^{Rounds} Profit_i \times Probability_i

STEP 6

The profit for each round is the amount you bet, and the probability is the chance of winning that round. If you win in the i-th round, your profit is the amount you bet in that round minus the total amount you have bet in all previous rounds.
Profiti=Beti−∑j=0i−1BetjProfit_i = Bet_i - \sum_{j=0}^{i-1} Bet_jProbabilityi=0.5i+1Probability_i =0.5^{i+1}

STEP 7

The amount you bet in each round is double the amount you bet in the previous round, starting with a bet of $1 in the first round.
Beti=2iBet_i =2^i

STEP 8

Now, plug in the values for the bet and the probability in the formula for the expected profit.
Expected profit=∑i=07(2i−∑j=0i−12j)×0.5i+1Expected\, profit = \sum_{i=0}^{7} (2^i - \sum_{j=0}^{i-1}2^j) \times0.5^{i+1}

STEP 9

implify the formula for the expected profit.
Expected profit=∑i=72i×.5i+−∑i=7∑j=i−2j×.5i+Expected\, profit = \sum_{i=}^{7}2^i \times.5^{i+} - \sum_{i=}^{7} \sum_{j=}^{i-}2^j \times.5^{i+}

STEP 10

Calculate the expected profit.
Expected profit=$0.50Expected\, profit = \$0.50The expected profit is $0.50.

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