5. The probabilities that the lifetime of a randomly chosen device is below average, average, above average are respectively 0.2, 0.5, and 0.3. The probability that the 3 randomly chosen devices lifetimes of 2 are above average and 1 below average equals

Choose one answer
a. 0.054
b. 0.018
c. 0.036
d. 0.012

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.054

Question 1 of 1:

Finding Standard Deviation
A simple random sample of FICO credit rating scores is listed below:

714 751 664 789 818 779 698 836 753 834 693 802
Find range, variance, and standard deviation.
As of this writing, the mean FICO score was reported to be 678. Based on these results, is a FICO score of 500 unusual? Why or why not?

To solve this problem, we can use the concept of probability.

Given:
- The probability that the lifetime of a randomly chosen device is below average = 0.2
- The probability that the lifetime of a randomly chosen device is average = 0.5
- The probability that the lifetime of a randomly chosen device is above average = 0.3

We need to find the probability that out of 3 randomly chosen devices, 2 have lifetimes above average and 1 has a lifetime below average.

To calculate this probability, we need to use the binomial distribution formula.

The binomial distribution formula is given by:
P(X=k) = nCr * p^k * (1-p)^(n-k)

Where:
- P(X=k) is the probability of getting k successes
- nCr is the notation for binomial coefficient and it is calculated as n! / (k! * (n-k)!)
- p is the probability of success for each trial
- k is the number of successful trials
- n is the total number of trials

In this case, we have n = 3, p = 0.3, k = 2 (above average) and 1 (below average).

Let's calculate the probability now:

P(X=2) = 3C2 * 0.3^2 * (1-0.3)^(3-2)
= 3 * 0.09 * 0.7
= 0.189

P(X=1) = 3C1 * 0.2^1 * (1-0.2)^(3-1)
= 3 * 0.2 * 0.64
= 0.384

Now, we need to calculate the combined probability of having 2 above average and 1 below average:

P(X=2 and X=1) = P(X=2) * P(X=1)
= 0.189 * 0.384
= 0.072576

Therefore, the probability that the 3 randomly chosen devices have 2 lifetimes above average and 1 below average is approximately 0.072576.

Now let's check the options given:

a. 0.054
b. 0.018
c. 0.036
d. 0.012

None of the options match our calculated probability.

Hence, none of the given options is correct.