A distribution center receives shipment of a product from three different factories in the quantities of 60, 35 and 25. Three times a product is selected at random, each time without replacement. Find the probability that (a) all three products came from the second factory and (b) none of the three products came from the second factory
(a) The probability that all products came from the second factory is
(b) The probability that none of the three products came from the second factory is
Part A: add all the numbers 60+35+25=120
Then divide...taking one away each time.
(35/120)(34/119)(33/118)=0.023
*Final Answer: 0.023
Part B: Add all the number together: 60+35+25=120
then subtract the second factory: 120-35=85
Then do same as part A
(85/120)(84/119)(83/118)=0.352
*Final answer: 0.352
(a) Well, if all three products came from the second factory, that means we need to select all three products from the second factory without replacement. The first time we pick a product, we have a 35 out of 120 chance (since there are 35 products from the second factory out of a total of 120 products). After picking one product, there are now 34 products from the second factory out of 119 total products. So the probability of picking a second product from the second factory is 34 out of 119. Finally, after picking two products, there are 33 products from the second factory out of 118 total products. Therefore, the probability of picking a third product from the second factory is 33 out of 118.
So, to find the probability that all three products came from the second factory, we multiply these individual probabilities together:
(35/120) * (34/119) * (33/118) = 0.0744
Therefore, the probability that all three products came from the second factory is 0.0744.
(b) Now, let's find the probability that none of the three products came from the second factory. This means that we need to select all three products from the first and third factories.
For the first product, we have 60 products in the first factory out of a total of 120, so the probability of selecting a product from the first factory is 60 out of 120.
After picking one product from the first factory, there are now 59 products left in the first factory out of 119 total products. So the probability of picking a second product from the first factory is 59 out of 119.
Finally, after picking two products, there are 25 products in the third factory out of 118 total products. Therefore, the probability of picking a third product from the third factory is 25 out of 118.
So, to find the probability that none of the three products came from the second factory, we multiply these individual probabilities together:
(60/120) * (59/119) * (25/118) = 0.0870
Therefore, the probability that none of the three products came from the second factory is 0.0870.
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.
To find the probabilities, we first need to determine the total number of ways to select three products from the three factories.
The total number of ways to select three products without replacement can be found using combinations. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items being selected. In this case, we have 120 products in total (60 + 35 + 25), and we want to select 3 products.
So, the total number of ways to select three products without replacement is:
nCr = 120! / (3!(120-3)!) = 120! / (3!117!) = (120 * 119 * 118) / (3 * 2 * 1) = 114,960.
Now we can determine the probabilities:
(a) The probability that all three products came from the second factory:
Since all three products are selected from the second factory, the number of ways to select them is 35C3. So, the probability is:
P(all from second factory) = 35C3 / 114,960.
(b) The probability that none of the three products came from the second factory:
Since none of the products are selected from the second factory, the number of ways to select them is (60 + 25)C3. So, the probability is:
P(none from second factory) = (60 + 25)C3 / 114,960.
Now you can calculate the two probabilities using these formulas.