In the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2,..., 36. To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots. If the number of the slot the ball falls into matches the number you selected, you win $35; otherwise you lose $1. Complete parts (a) through (g) below.
A)Construct a probability distribution for the random variable X, the winnings of each spin.
X P(X)
35 .0263
-1 .9737
(b) Determine the mean and standard deviation of the random variable X. Round your results to the nearest penny.
can you explain how they came up with the mean of -.05 and standard deviation of 5.76
The mean is calculated by the equation =(35)(.0263)+(-1)(.9737)
Perfect!
To calculate the mean and standard deviation of a probability distribution, we need to use the formula:
Mean (μ) = Σ (X * P(X))
Standard Deviation (σ) = √{Σ [(X - μ)^2 * P(X)]}
For the given probability distribution:
X P(X)
35 .0263
-1 .9737
Multiply each value of X by its corresponding probability P(X) and calculate the sum:
(35 * 0.0263) + (-1 * 0.9737) = 0.921 + (-0.9737) = -0.0527
The mean of the random variable X is approximately -0.05 (rounded to the nearest penny).
To calculate the standard deviation, we need to find the squared difference between each value of X and the mean, multiply it by its corresponding probability P(X), and calculate the sum:
[((35 - (-0.05))^2 * 0.0263) + ((-1 - (-0.05))^2 * 0.9737)]^0.5
= [((35.05)^2 * 0.0263) + ((-0.95)^2 * 0.9737)]^0.5
= [(1226.702025 * 0.0263) + (0.9025 * 0.9737)]^0.5
= [32.23686283475 + 0.87872545]^0.5
= 33.11558828475^0.5
= 5.7629244
The standard deviation of the random variable X is approximately 5.76 (rounded to the nearest penny).
To find the mean of a probability distribution, you multiply each value of the random variable by its corresponding probability and then sum them up. In this case, the possible values of X are 35 (winning $35) with a probability of 0.0263, and -1 (losing $1) with a probability of 0.9737.
Mean (μ) = (35 * 0.0263) + (-1 * 0.9737)
= 0.92105 - 0.9737
= -0.05265
So, the mean of the random variable X is approximately -0.05 (rounded to the nearest penny).
To find the standard deviation of a probability distribution, you need to calculate the variance first. The formula for variance is the square of the difference between each value and the mean, multiplied by their corresponding probabilities, summed up.
Variance (σ²) = [(35 - (-0.05))² * 0.0263] + [(-1 - (-0.05))² * 0.9737]
= [(35.05)² * 0.0263] + [(-0.95)² * 0.9737]
= [1225.5025 * 0.0263] + [0.9025 * 0.9737]
= 32.21076575 + 0.878453725
= 33.089219475
Once you have the variance, you can find the standard deviation (σ) by taking the square root of the variance.
Standard Deviation (σ) = √(33.089219475)
= 5.759
So, the standard deviation of the random variable X is approximately 5.76 (rounded to the nearest penny).