Math Snap

STEP 1

Assumptions1. The initial bet is $1.
. If you lose, you double your bet.
3. You continue this strategy until you win a toss.
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.
$$Rounds = \lfloor \log2(Bankroll) \rfloor$$

STEP 3

Plug in the given value for the bankroll to calculate the maximum number of rounds.
$$Rounds = \lfloor \log2(127) \rfloor$$

STEP 4

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

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.
$$Expected\, profit = \sum_{i=0}^{Rounds} Profit_i \times Probability_i$$

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.
$$Profit_i = Bet_i - \sum_{j=0}^{i-1} Bet_j$$$$Probability_i =0.5^{i+1}$$

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.
$$Bet_i =2^i$$

STEP 8

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

STEP 9

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

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