Solving a Probability Problem: Purchasing Defective TVs for a Hotel

[leright]

Any help would be appreciated. I just want a push in the right direction...I am kinda struggling with these problems.

A shipment of 12 TVs contains 3 defective sets. In how many ways can a hotel purchase 5 of these sets and receive at least 2 of the defective sets?

Thanks.

 

[Muzza]

Count the number of sets which can be written like so:

{all three defective TVs} $$\cup$$ {a set containing precisely two working TVs}, or

{a set containing precisely two defective TVs} $$\cup$$ {a set containing precisely three working TVs}.

 

[quicknote]

I understand that probability problems can be difficult to solve, but with the right approach, they can be tackled successfully. In this case, we can use the concept of combinations to solve this problem.

First, let's identify the total number of ways to choose 5 TVs from the 12 available. This can be calculated using the combination formula, nCr, where n is the total number of items and r is the number of items to be chosen. In this case, n=12 and r=5, so the total number of ways to choose 5 TVs is 12C5, which is equal to 792.

Next, we need to determine the number of ways to choose at least 2 defective sets from the 3 available. This can be done by calculating the combinations of choosing 2, 3, 4, or 5 defective sets from the total of 3. This can be represented as 3C2 + 3C3 + 3C4 + 3C5, which equals 3 + 1 + 0 + 0, or 4.

Finally, we can calculate the number of ways to choose 5 TVs and receive at least 2 defective sets by multiplying the two values we calculated above. This gives us a total of 792*4 = 3168 possible ways.

I hope this helps guide you in the right direction. Remember, when solving probability problems, it is important to clearly define the problem, identify the relevant variables, and use appropriate formulas or methods to find the solution. Keep practicing and you will become more confident in solving these types of problems. Good luck!