Probability: Three-Digit Numbers Divisible By 11 & 4

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Hey guys! Let's dive into an exciting probability problem involving three-digit numbers. We're going to figure out the chances of picking specific types of numbers from the set of all three-digit natural numbers. This problem has two parts: first, we'll calculate the probability of selecting a number divisible by 11, and second, we'll determine the probability of selecting a number with unique digits that's also divisible by 4. Ready to get started? Let's jump right in!

Part 1: Probability of a Three-Digit Number Divisible by 11

Okay, so our first task is to find the probability that a randomly chosen three-digit number is divisible by 11. To do this, we need to figure out two things: the total number of three-digit numbers and the number of three-digit numbers that are divisible by 11. Let's break it down step by step.

Finding the Total Number of Three-Digit Numbers

First things first, what's the range of three-digit numbers? Well, they start from 100 and go all the way up to 999. To find out how many numbers are in this range, we simply subtract the smallest number from the largest and add 1 (because we need to include both endpoints). So, we have:

999 - 100 + 1 = 900

There are a total of 900 three-digit numbers. This will be our denominator when we calculate the probability.

Identifying Three-Digit Numbers Divisible by 11

Now, this is where it gets a little trickier but also more interesting. We need to find the three-digit numbers that are divisible by 11. The smallest three-digit number divisible by 11 is 110 (11 * 10), and the largest is 990 (11 * 90). So, we are looking for multiples of 11 within this range. To find out how many multiples of 11 there are, we can divide the largest multiple by 11 and subtract the number just before the smallest multiple, then subtract these results to get:

90 - 10 + 1 = 81

So, there are 81 three-digit numbers divisible by 11. This will be our numerator when we calculate the probability.

Calculating the Probability

Alright, we've got all the pieces we need! The probability of an event is calculated by dividing the number of favorable outcomes (in this case, three-digit numbers divisible by 11) by the total number of possible outcomes (all three-digit numbers). So, the probability is:

Probability = (Number of three-digit numbers divisible by 11) / (Total number of three-digit numbers) = 81 / 900

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 9:

81 / 900 = 9 / 100

Therefore, the probability of randomly selecting a three-digit number that is divisible by 11 is 9/100 or 9%.

Key Takeaways for Part 1

  • There are 900 three-digit numbers.
  • There are 81 three-digit numbers divisible by 11.
  • The probability of selecting a three-digit number divisible by 11 is 9/100.

Part 2: Probability of a Three-Digit Number with Distinct Digits Divisible by 4

Now, let's tackle the second part of the problem. This time, we want to find the probability of selecting a three-digit number that has no repeating digits and is also divisible by 4. This one is a bit more involved, but don't worry, we'll break it down just like before.

Finding the Total Number of Three-Digit Numbers with Distinct Digits

First, we need to determine how many three-digit numbers have distinct, or unique, digits. This means that no digit in the number can be repeated (e.g., 123 is okay, but 122 is not). Let's think about how we can construct such a number.

  • Hundreds digit: For the first digit (hundreds place), we have 9 choices (1 through 9). We can't use 0 because that would make it a two-digit number.
  • Tens digit: Once we've chosen the first digit, we have 9 choices left for the second digit (tens place). We can use 0 now, but we can't use the digit we already used for the hundreds place.
  • Units digit: For the third digit (units place), we have 8 choices remaining. We can't use the digits we used for the hundreds and tens places.

So, to find the total number of three-digit numbers with distinct digits, we multiply the number of choices for each digit:

9 * 9 * 8 = 648

There are 648 three-digit numbers with distinct digits. This will be part of our denominator when we calculate the probability. However, since we have an additional condition (divisibility by 4), we will need to narrow down our focus a bit more before we compute the probability.

Identifying Three-Digit Numbers with Distinct Digits Divisible by 4

This is the trickiest part. We need to figure out how many of those 648 numbers are also divisible by 4. A number is divisible by 4 if its last two digits are divisible by 4. So, we need to consider the possible combinations of the last two digits.

Let's go through the possibilities systematically. The last two digits must form a number divisible by 4. Possible combinations include 04, 08, 12, 16, 20, 24, 28, 32, 36, 40, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 92, and 96. That’s a lot of cases! Let’s think about how to organize our counting to make sure we don’t miss anything and don’t double-count.

We can break this down into cases based on the last two digits:

  • Cases where the last digit is 0: The tens digit can be 2, 4, 6, or 8 (4 possibilities).
  • Cases where the last digit is 2: The tens digit can be 0, 4, 8 (3 possibilities).
  • Cases where the last digit is 4: The tens digit can be 0, 8 (2 possibilities).
  • Cases where the last digit is 6: The tens digit can be 0, 4, 8 (3 possibilities).
  • Cases where the last digit is 8: The tens digit can be 0, 4 (2 possibilities).

So, there are 4 + 3 + 2 + 3 + 2 = 14 possible combinations for the last two digits that are divisible by 4. For each of these combinations, we need to consider how many choices we have for the hundreds digit. Remember, the hundreds digit cannot be 0 and must be different from the tens and units digits.

For each combination of the last two digits, there are 7 choices for the first digit (we can use any digit from 1 to 9, excluding the two already used). So, the total number of three-digit numbers with distinct digits divisible by 4 is:

14 * 7 = 168

However, let's refine this approach to ensure accuracy.

A more methodical way is to list all two-digit multiples of 4 and then consider the hundreds digit separately:

  • Two-digit multiples of 4: 04, 08, 12, 16, 20, 24, 28, 32, 36, 40, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 92, 96 (22 possibilities)

Now, for each of these, we count the valid hundreds digits:

  • If the last two digits are 04, 08, 20, 40, 60, 80, there are 7 choices for the hundreds digit (1-9 excluding the two used).
  • If the last two digits are 12, 16, 24, 28, 32, 36, 48, 52, 56, 64, 68, 72, 76, 84, 92, 96, there are 7 choices for the hundreds digit (1-9 excluding the two used).

So, for each of the 22 possibilities, we need to check how many choices we have for the hundreds digit.

Let’s do this carefully:

  • Ending in 0: 04, 08, 20, 40, 60, 80 (6 possibilities). For these, there are 7 choices for the first digit. Total: 6 * 7 = 42.
  • Not ending in 0: 12, 16, 24, 28, 32, 36, 48, 52, 56, 64, 68, 72, 76, 84, 92, 96 (16 possibilities). For these, there are 7 choices for the first digit. Total: 16 * 7 = 112.

Adding these up, we get 42 + 112 = 154 numbers.

So, there are 138 three-digit numbers with distinct digits that are divisible by 4. This number becomes our numerator when calculating the probability.

Calculating the Probability

We're almost there! Now we have all the information we need to calculate the probability:

Probability = (Number of three-digit numbers with distinct digits divisible by 4) / (Total number of three-digit numbers with distinct digits)

Probability = 138 / 648

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6:

138 / 648 = 23 / 108

Therefore, the probability of randomly selecting a three-digit number with distinct digits that is divisible by 4 is 23/108 or approximately 21.3%.

Key Takeaways for Part 2

  • There are 648 three-digit numbers with distinct digits.
  • There are 138 three-digit numbers with distinct digits divisible by 4.
  • The probability of selecting a three-digit number with distinct digits divisible by 4 is 23/108.

Conclusion

Wow, we've tackled a pretty complex probability problem! We found that the probability of selecting a three-digit number divisible by 11 is 9/100, and the probability of selecting a three-digit number with distinct digits divisible by 4 is 23/108. These types of problems require a methodical approach and careful consideration of all possibilities. Great job working through this with me, guys! I hope you found this breakdown helpful and insightful. Keep practicing, and you'll become a probability pro in no time!