MIS 570-DQ 2
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3.1
Here is given,
Win: $35 with probability 1/38
Loss: -$1 with probability 37/38
Single Bet:
Take a single roulette bet situation in which a $1 wager is put on a single number. The expected value, or mean, is determined as -0.05. This indicates that, on average, the outcome results in a 5-cent loss per wager. Concurrently, the standard deviation, a measure of the dispersion of conceivable outcomes, approaches $0.95.
Mean (µ) Calculation: µ= (35) * (1/38) – (1) * (37/38)
µ= 35/38- 37/38
µ ≈ – 0.0526
100 Bets:
Expanding this research to include 100 similar bets, the average net gain is -$5.26, which is calculated by multiplying 100 by the average of a single bet. Similarly, the standard deviation of net profits over 100 bets is $9.46, calculated as the square root of 100 times the standard deviation of a single bet.
µ
of 100 beats = 100 * µ of single beat
µ
of 100 beats ≈ 100 * -0.0526 µ
of 100 beats ≈ 100 * -0.0526
µ
of 100 beats ≈ - 5.26
Alternatively, the standard deviation of net winnings over 100 bets is computed as follows: The standard deviation of 100 bets equals 100 times the standard deviation of a single bet. The standard deviation of 100 bets is equal to 100 times the standard deviation of a single bet. The standard deviation of 100 bets is around 10 x 0.95. The standard deviation of 100 bets is around 10×0.95. The standard deviation of 100 bets is approximately 9.46. The standard deviation of 100 bets is approximately 9.46.
95% Confidence Interval:
Formula= Mean ± 1.96 * Standard Deviation.
A confidence interval is used to estimate the expected range of net profits from 100 bets with a 95% confidence level. This interval, using the mean and standard deviation, has an estimated range of -$24.88 to $14.36. Thus, with a 95% confidence level, it is expected that net earnings from 100 bets will fall within this range.
Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2016). Introduction to Probability and Statistics.
Cengage Learning.
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