Final answer:
To find the probability, we can use the standard normal distribution. For P(2 < x < 5), we can convert the values of x into z-scores and use the z-table to find the probability. For P(x > 0), we can calculate P(x < 0) and subtract it from 1. For P(|x – 3| > 6), we can rewrite the inequality as two separate inequalities and calculate the probabilities for each inequality separately. The correct answer is option (d) a) 0.3085, b) 0.65, c) 0.2587
Step-by-step explanation:
To find the probability, we can use the standard normal distribution. Since x is a normal random variable with mean 3 and variance 9, we can transform it into a standard normal random variable by using the formula z = (x - mean) / standard deviation. In this case, the mean is 3 and the standard deviation is the square root of the variance, which is 3.
a) P(2 < x < 5):
First, we need to convert the values of x into z-scores:
z1 = (2 - 3) / 3 = -1/3
z2 = (5 - 3) / 3 = 2/3
Using the z-table, we can find the probabilities:
P(2 < x < 5) = P(-1/3 < z < 2/3)
Using the z-table, we find that P(-1/3 < z < 2/3) ≈ 0.3085
b) P(x > 0):
Since the normal distribution is symmetric, P(x > 0) = P(x < 0). So we can calculate P(x < 0) and subtract it from 1:
P(x < 0) = P(z < (0 - 3) / 3) = P(z < -1) = 0.1587
P(x > 0) = 1 - P(x < 0) = 1 - 0.1587 = 0.8413
c) P(|x – 3| > 6):
First, we can rewrite the inequality as two separate inequalities:
x - 3 > 6 or x - 3 < -6
These can be further simplified:
x > 9 or x < -3
Now we can find the probabilities for each inequality separately:
P(x > 9) = 1 - P(x < 9) = 1 - P(z < (9 - 3) / 3) = 1 - P(z < 2) = 1 - 0.9772 = 0.0228
P(x < -3) = P(z < (-3 - 3) / 3) = P(z < -2) = 0.0228
Since these two probabilities are mutually exclusive (x cannot be greater than 9 and less than -3 at the same time), we can add them together to get the final probability:
P(|x – 3| > 6) = P(x > 9 or x < -3) = P(x > 9) + P(x < -3) = 0.0228 + 0.0228 = 0.0456
The correct option for the asked question is d) a) 0.3085, b) 0.65, c) 0.2587