A gambling game pays 4 to 1 and the chance of winning is 1 in 6. Suppose you bet $1 on this game 600 times independently.
3A - Find the expected number of times you win.
3B - Find the SE of the number of times you win.
3C - Find the chance that you lose more than $50 (that is, your net gain in the 600 bets is less than -$50).
a) 100
135
What is the 135
To find the answers to these questions, we need to understand the concept of expected value, standard error, and probability. Let's break down each question and explain how to solve it.
3A - Find the expected number of times you win:
The expected value is the average outcome we can expect over the long run. In this case, we expect to win with a probability of 1/6, and when we win, we receive $4. Therefore, the expected number of times you win can be calculated as follows:
Expected Number of Wins = (Probability of Winning) * (Number of Bets) = (1/6) * (600) = 100
So, the expected number of times you win is 100.
3B - Find the standard error (SE) of the number of times you win:
The standard error measures the variation or uncertainty around the expected value. It is calculated using the following formula:
Standard Error = sqrt(Number of Bets * Probability of Winning * Probability of Losing)
Probability of Losing = 1 - Probability of Winning = 1 - (1/6) = 5/6
Using this formula, we can calculate the standard error as follows:
Standard Error = sqrt(600 * (1/6) * (5/6)) ≈ 8.49
So, the standard error of the number of times you win is approximately 8.49.
3C - Find the chance that you lose more than $50:
To calculate the probability that you lose more than $50, we need to consider the net gain or loss from the 600 bets. Each win gives you a profit of $4, and each loss results in a loss of $1. The net gain in the 600 bets can be represented by a binomial distribution.
The probability of a single bet resulting in a loss is 5/6, and the probability of a win is 1/6. We need to figure out the probability of having a net loss greater than $50 in 600 bets.
To calculate this probability, we can use statistical software or online calculators. Alternatively, we can approximate this probability by applying the normal approximation to the binomial distribution.
By approximating the binomial distribution to a normal distribution, we can calculate the mean (μ) and standard deviation (σ) for the net gain or loss.
Mean (μ) = Number of Bets * (Probability of Winning * Win Amount - Probability of Losing * Loss Amount)
= 600 * ((1/6) * 4 - (5/6) * 1) = 600 * (2/3 - 5/6) = -100
Standard Deviation (σ) = sqrt(Number of Bets * Probability of Winning * Probability of Losing * (Win Amount - Loss Amount)^2)
= sqrt(600 * (1/6) * (5/6) * (4 - 1)^2)
= sqrt(600 * (1/6) * (5/6) * 9)
= sqrt(225) = 15
Using these values, we can calculate the standard score (z-score) for a net loss of $50:
z-score = (Net Loss - Mean) / Standard Deviation
= (-50 - (-100)) / 15
= 50 / 15 ≈ 3.33
Next, we can use a standard normal distribution table or a calculator to find the probability associated with a z-score of 3.33. This probability represents the chance of losing more than $50.
Keep in mind that this approximation assumes that the number of bets is large enough for the normal approximation to be valid.