Question

The Blacktop Speedway is a supplier of automotive parts. Included in stock are 10 speedometers that are correctly calibrated and two that are not. Three speedometers are randomly selected without replacement. Let the random variable

x
represent the number that are not correctly calibrated.

Complete the probability distribution table. (Report probabilities accurate to 4 decimal places.)
x. P(x)
0
1
2
3


Find the mean of this probability distribution. (Appropriate rounding rules apply.)
mean =

Answer 1

To complete the probability distribution for the number of uncalibrated speedometers, we calculate the probabilities of getting 0, 1, 2, or 3 uncalibrated speedometers when 3 speedometers are randomly selected. The mean of the probability distribution is 0.7273.

Step-by-step explanation:

To complete the probability distribution table, we need to find the probability of getting 0, 1, 2, or 3 speedometers that are not correctly calibrated when 3 speedometers are randomly selected without replacement from a stock that includes 10 calibrated speedometers and 2 uncalibrated speedometers.

P(x = 0) = (10/12) * (9/11) * (8/10) = 0.5455

P(x = 1) = (2/12) * (10/11) * (9/10) + (10/12) * (2/11) * (9/10) + (10/12) * (9/11) * (2/10) = 0.4182

P(x = 2) = (2/12) * (1/11) * (10/10) + (2/12) * (10/11) * (1/10) + (10/12) * (2/11) * (1/10) = 0.0364

P(x = 3) = (2/12) * (1/11) * (0/10) = 0.0000

The mean of this probability distribution is found by multiplying each value of x by its corresponding probability and then summing them up. mean = 0 * 0.5455 + 1 * 0.4182 + 2 * 0.0364 + 3 * 0.0000 = 0.7273

Answer 2

0.4

Step-by-step explanation:

To complete the probability distribution table, we need to determine the probability of selecting each possible value of the random variable x.

Since there are 10 correctly calibrated speedometers and 2 that are not, there are a total of 12 speedometers, and the probability of selecting a speedometer that is not correctly calibrated on the first draw is 2/12 = 1/6. After the first draw, there are 11 speedometers remaining, including one that is not calibrated, so the probability of selecting a second speedometer that is not calibrated is 1/11. Finally, on the third draw, there are 10 speedometers remaining, including the one that is not calibrated, so the probability of selecting a third speedometer that is not calibrated is 1/10.

Using these probabilities, we can complete the probability distribution table as follows:

x P(x)

0 (10/12) * (9/11) * (8/10) = 0.60

1 3 * (1/6) * (5/11) * (8/10) = 0.36

2 3 * (1/6) * (1/11) * (8/10) = 0.02

3 (1/6) * (1/11) * (2/10) = 0.00

To find the mean of this probability distribution, we can use the formula:

mean = Σ(x * P(x))

where Σ denotes a sum over all possible values of x.

Using the values from the probability distribution table, we have:

mean = (0 * 0.60) + (1 * 0.36) + (2 * 0.02) + (3 * 0.00) = 0.40

Therefore, the mean of this probability distribution is 0.40.