56. Defective Disks A pack of 100 recordable DVDs contains 5 d...

Question

56. Defective Disks A pack of 100 recordable DVDs contains 5 defective disks. You select four disks. What is the probability of selecting at least three non defective disks?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A shipment of 120 smartphone batteries contains 8 that are faulty. If a quality inspector randomly selects 6 batteries, what is the probability that at least 5 of them are not faulty? Awesome. So it appears for this particular prompt, we're ultimately trying to determine that if an inspector randomly selects 6 batteries from this shipment, what would be the probability that at least 5 of them are not faulty? So with that in mind, now that we know what we're trying to solve for, let us read off our multiple choice answers to see what our final answer might be. A is 0.7548. B is 0.8614. C is 0.8923 and D is 0.9488. Awesome. So, first off, we need to recall and note there are 120 batteries total, which I'll say T for total, with 8 batteries that are faulty, which I'll just simply call F. And there are 112 that are non-faulty, which I'll call NF for short. Our second step now is we need to recall and note that we want to determine the probability that at least 5 out of the 6 selected batteries are not faulty. And this will include the cases where 5 or 6 are non-faulty. So this will mean that exactly 5 of them are considered good and 1 is faulty, or 6 of them are considered good and 0 of them are considered faulty. Our 3rd step. As we need to determine the total number of ways to choose 6 batteries from 120. Which we could use our combination formula, as we should recall, which we could write a fancy way of writing a combination formula based prom is as a binomial coefficient. So we could write this as 120 chooses 6, and once again, the total number of ways to choose 6 batteries from 120, we could write this as 120 chooses 6, which when we use our combination formula. And we plug that into our calculator. We will find that it's ultimately equal to 3,652,745, 460. So hundreds, thousands, millions, billions, so it's 3 billion. 652,745,460, different ways. So, it might not work in your calculator, so you might have to use online software or break it up into parts when you're using your combination formula. So moving right along here, our fourth step is we need to determine the number of ways to choose 6 non-faulty batteries, which we could write this as 112 chooses 6, which when we plug these values into our combination formula, we will get 2,392,407, 864. So, hundreds, thousands, millions, billions, so we'll get 2392407,864 different ways. So our 5th step now is we need to determine the number of ways to choose 5 good and 1 faulty battery is. 112 chooses 5 multiplied by 8 chooses 1. Which is equal to 134. Million 153. 1,712. So once again, 134, 153,712 multiplied by 8, which will give us an answer of 1 billion. 73229,696. Fantastic. Moving right along here, our 6th step is we need to add the results from step 4 and 5 to get the total number of favorable outcomes. So the total number of favorable outcomes, which you'll call TFO for short. So the total number of favorable outcomes is equal to 2392407,864 plus 173229,696, which will be all equal to $3 billion when we add them 3 billion. 465,637,560. Awesome. So now our seventh and final step is we need to determine the probability, and the probability P is equal to the number of favorable outcomes which is equal to 3 billion. 465,637,560 divided by the total number of outcomes, which is 3. 652,745,460 Which when we plug this into our calculator and we round to 4 decimal places, we will find that it's equal to 0.9488, and that's it. That is our final answer. Hooray, we did it. And this corresponds to multiple choice answer letter D, which once again is 0.9488. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

56. Defective Disks A pack of 100 recordable DVDs contains 5 defective disks. You select four disks. What is the probability of selecting at least three non defective disks?

Key Concepts

Probability

Combinations

Complement Rule

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