66. Access Code An access code consists of six characters. For...

Question

66. Access Code An access code consists of six characters. For each character, any letter or number can be used, with the exceptions that the first character cannot be 0 and the last two characters must be odd numbers.
a. What is the probability of randomly selecting the correct access code on the first try?

a. What is the probability of randomly selecting the correct access code on the first try?

Accepted Answer

Hello there. Today we're going to 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 security badge code is made up of 5 characters. Each character can be any uppercase letter from A to Z, or any digit from 0 to 9. Except that the first character must be a letter, and the last character must be an even digit. What is the probability of guessing the correct badge code on the first attempt? Awesome. So it appears for this particular problem, we're ultimately trying to determine what is the probability of guessing the correct badge code on the first attempt. So with that in mind, now that we know that we're ultimately trying to determine this probability of guessing the correct badge code on the first attempt of guessing it, let's read off our multiple choice answers to see what our final answer might be. A is 1 divided by 4,852,224. B is 1 divided by 665,280, C is 1 divided by 1,679,616. And finally, D is 1 divided by 299,520. Awesome. So, with that in mind, Our first step to solve this problem is we need to determine the number of possible codes. So for the first character, as we should note, it must be an uppercase letter, so it needs to be a letter that's either A or rather capital A to Z. So that means, as we should recall, there is 26 letters in the alphabet, so the number of choices will be 26. And for the 2nd, Third, And 4th characters. We need to note that each can be any uppercase letter, which is A to Z, specifically capital A to Z, but in an effort to save time, I'll just say A to Z, but note that A to Z are all capital letters for this particular prompt. So once again, for the 2nd, 3rd and 4th characters, each can be any uppercase letter A to Z. Or any digit from 0 to 9. Fantastic. So let us note that our total number of letters is 26 and our total number of digits is 10. So the total number of choices, so the total choices for each position is going to be equal to 26 plus 10, which is going to be equal to 36 total. And as for our 5th and final character, the last character of our badge number, it has to be an even number, so I'll put a letter E for even. So that means we need to recall that our even digits that lie within our specific range for our digits are 0.2.4.8. Fantastic and don't forget that it's 024, 6 and 8. Fantastic. So that means our total number of choices for even digits is 5. So, it's important to recall and note that the fundamental counting principle will be used for this particular problem will be used here. And as we should recall, the fundamental counting principle states that if there are ways to do one thing and N ways to do another, then there are M multiplied N ways to do both. So the total number of possible codes is 26 multiplied by 36 multiplied by 36 multiplied by 36 multiplied by 5, which is gonna be equal to 6 million. 65,280. Awesome. Our second step now is we need to determine the probability of guessing correctly. So only one code is correct, so the probability of guessing the correct code on the first try. Undoubtedly will have to be So once again, we have to recall and note that the probability, because you're only trying to figure out the probability of guessing the correct badge code on the first try will have to be that the probability P. is equal to 1 divided by 6 million. 65,000. 280 and that is our final answer. Hooray, we did it, and this corresponds with multiple choice answer letter B, which once again is 1 divided by 665,280. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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Key Concepts

Combinatorics

Probability

Permutations and Restrictions

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