To calculate the mean value of x, we multiply each value of x by its corresponding probability and sum them up:

Mean = (0)(0.05) + (1)(0.10) + (2)(0.20) + (3)(0.35) + (4)(0.20) + (5)(0.08) + (6)(0.02)
= 0 + 0.10 + 0.40 + 1.05 + 0.80 + 0.40 + 0.12
= 2.87

To calculate the standard deviation of x, we first need to calculate the variance. The variance of x is calculated as the sum of [(x - mean)^2 * p(x)]:

Variance = [(0 - 2.87)^2 * 0.05] + [(1 - 2.87)^2 * 0.10] + [(2 - 2.87)^2 * 0.20] + [(3 - 2.87)^2 * 0.35] + [(4 - 2.87)^2 * 0.20] + [(5 - 2.87)^2 * 0.08] + [(6 - 2.87)^2 * 0.02]
= [(-2.87)^2 * 0.05] + [(-1.87)^2 * 0.10] + [(-0.87)^2 * 0.20] + [(0.13)^2 * 0.35] + [(1.13)^2 * 0.20] + [(2.13)^2 * 0.08] + [(3.13)^2 * 0.02]
= [8.2769 * 0.05] + [3.4969 * 0.10] + [0.7569 * 0.20] + [0.0169 * 0.35] + [1.2769 * 0.20] + [4.5169 * 0.08] + [9.7569 * 0.02]
= 0.4138 + 0.3497 + 0.1514 + 0.0059 + 0.2554 + 0.3614 + 0.1951
= 1.7337

The standard deviation of x is the square root of the variance:

Standard deviation = sqrt(1.7337)
= 1.3166 (rounded to four decimal places)

To calculate the probability that the number of lines in use is farther than 3 standard deviations from the mean value, we need to calculate the probability of x being less than mean - 3*standard deviation or greater than mean + 3*standard deviation.

Probability = P(x < 2.87 - 3*1.3166) + P(x > 2.87 + 3*1.3166)
= P(x < -2.0666) + P(x > 7.3033)

Since both of these x values are outside the range of the probability distribution, the probability is 0.

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