Question

We are at a casino playing a game. the bet is always $1. each time we win, we win $1, and each time we loss, we forfeit the $1 bet. the winning probability is 50% and we take $10000 to the casino tonight.o. how much do we have after 10 rounds? what is the probability that we make positive profits from the 10 rounds (we have more than 10000 after 10 rounds). do a simulation with 500 iterations to answer.

Round ID Win/Loss Bet Next Round Money We Have After Round
Before Start 1 10000
1
2
3
4
5
6
7
8
9
10

Answer 1

To calculate the amount after 10 rounds, consider that there are 2 to the power of 10 possible outcomes with independent bets. Use the binomial distribution formula to calculate the probability of making a profit. A 500-iteration simulation would model the game outcomes to provide an estimated probability.

Step-by-step explanation:

In a casino game with a 50% winning probability, a bet of $1 is made, and the player starts with $10000.

After 10 rounds, the amount of money the player has can vary depending on whether they win or lose each round. Since each bet is independent, there are 210 (1024) possible outcomes after 10 rounds. For positive profits after 10 rounds, the player needs to win more than half of the rounds.

To calculate this, we identify the outcomes where the number of wins exceeds 5. The probability of exactly k wins in 10 rounds can be calculated using the binomial distribution formula P(X = k) = C(10, k) 0.5k 0.510-k, where C(10, k) represents the binomial coefficient or the number of combinations of k wins out of 10.

The simulation with 500 iterations will involve using a programming environment or software to model 500 sequences of 10 rounds of the game and determine the frequency of sessions ending with more than $10000.

Each iteration would randomly assign a win or loss for each round following the 50% win probability, and record the ending balance