To solve these probability problems, we need to use the concepts of combinations and probability. Let's start with part (a).
(a) The probability that all three products came from the second factory:
Step 1: Calculate the total number of ways to select 3 products out of a total of 120 products (60 + 35 + 25):
Total ways = C(120, 3)
Step 2: Calculate the number of ways to select all 3 products from the second factory:
Ways to select from the second factory = C(35, 3)
Step 3: Calculate the probability using the formula:
Probability = Ways to select from the second factory / Total ways
Probability = C(35, 3) / C(120, 3)
Now let's solve it:
Total ways = C(120, 3) = (120*119*118)/(3*2*1) = 266,280
Ways to select from the second factory = C(35, 3) = (35*34*33)/(3*2*1) = 5,365
Probability = 5,365 / 266,280 ≈ 0.0201
Therefore, the probability that all three products came from the second factory is approximately 0.0201.
Now let's move on to part (b).
(b) The probability that none of the three products came from the second factory:
Step 1: Calculate the total number of ways to select 3 products out of a total of 120 products:
Total ways = C(120, 3)
Step 2: Calculate the number of ways to select 3 products not from the second factory:
Ways to select not from the second factory = C(60+25, 3) = C(85, 3)
Step 3: Calculate the probability using the formula:
Probability = Ways to select not from the second factory / Total ways
Probability = C(85, 3) / C(120, 3)
Let's solve it:
Total ways = C(120, 3) = 266,280
Ways to select not from the second factory = C(85, 3) = (85*84*83)/(3*2*1) = 1,681,820
Probability = 1,681,820 / 266,280 ≈ 6.3156
Therefore, the probability that none of the three products came from the second factory is approximately 6.3156.