A number is chosen from the first 24 positive integers. Find the probability that:
a) the number is divisible by 3 given that it is divisible by 4
b) the number is divisible by 6 given that it is divisible by 3
Just need help with either one, thank you for your time.
For future students who google this:
A) 2/6 or 1/3
B) 1/2
12, 24
Q2. Numbers 1 - 24 that are divisible by 3: 3, 6, 9,12, 15,18, 21, 24
Which of those are also divisible by 6?
6, 12, 18, 24 So therefore 4/8 = 1/2 = Pr(d6Id3)
a) Ah, the divisibility game, I like it! So, let's see. Given that a number is divisible by 4, we know that it must be even. Out of the first 24 positive integers, there are 12 even numbers (2, 4, 6, ... 22, 24).
Now, let's find out how many of these even numbers are also divisible by 3. We have 4, 8, 12, 16, 20, and 24. That's a total of 6 numbers.
So, the probability that a number chosen is divisible by 3 given that it is divisible by 4 is 6/12, which simplifies to 1/2. Voila!
b) Now, given that a number is divisible by 3, we need to figure out how many of the first 24 positive integers satisfy this condition. Turns out, out of the first 24 positive integers, 8 of them are divisible by 3 (3, 6, 9, 12, 15, 18, 21, 24).
Out of these 8 numbers, we need to find how many are divisible by 6 (which is, in turn, divisible by 3). Luckily, every other number in our list (3, 9, 15, and 21) is also divisible by 6! So, that makes 4 numbers.
The probability that a number chosen is divisible by 6 given that it is divisible by 3 is 4/8, which can be simplified to 1/2. Ta-da!
Hope that added a dash of fun to your math journey! Let me know if you need any more chuckles or assistance with anything else.
Sure! Let's start with part a) of the question.
a) The number is divisible by 3 given that it is divisible by 4.
To solve this problem, we need to determine the count of numbers that are divisible by both 3 and 4, and then divide it by the total count of numbers divisible by 4.
Step 1: Count the numbers divisible by 4.
Out of the first 24 positive integers, we need to determine how many are divisible by 4. So, let's list them out: 4, 8, 12, 16, 20, 24. There are six numbers divisible by 4.
Step 2: Count the numbers divisible by both 3 and 4.
Now, let's figure out how many numbers are divisible by both 3 and 4. These numbers need to satisfy both divisibility conditions. So, we can observe that only 12 and 24 satisfy both conditions.
Step 3: Calculate the probability.
The probability is defined as the number of favorable outcomes (numbers divisible by both 3 and 4) divided by the total number of possible outcomes (numbers divisible by 4).
Probability (a) = Count of numbers divisible by both 3 and 4 / Count of numbers divisible by 4
So, Probability (a) = 2 / 6 (since there are two numbers divisible by both 3 and 4 and six numbers divisible by 4)
Simplifying, Probability (a) = 1 / 3
Therefore, the probability is 1/3 or approximately 0.333.
Now, let's move on to part b) of the question.
b) The number is divisible by 6 given that it is divisible by 3.
To solve this problem, we need to determine the count of numbers divisible by both 3 and 6, and then divide it by the total count of numbers divisible by 3.
Step 1: Count the numbers divisible by 3.
Out of the first 24 positive integers, we need to determine how many are divisible by 3. So, let's list them out: 3, 6, 9, 12, 15, 18, 21, 24. There are eight numbers divisible by 3.
Step 2: Count the numbers divisible by both 3 and 6.
Now, let's figure out how many numbers are divisible by both 3 and 6. These numbers need to satisfy both divisibility conditions. So, we can observe that 6, 12, 18, and 24 satisfy both conditions.
Step 3: Calculate the probability.
The probability is defined as the number of favorable outcomes (numbers divisible by both 3 and 6) divided by the total number of possible outcomes (numbers divisible by 3).
Probability (b) = Count of numbers divisible by both 3 and 6 / Count of numbers divisible by 3.
So, Probability (b) = 4 / 8 (since there are four numbers divisible by both 3 and 6 and eight numbers divisible by 3)
Simplifying, Probability (b) = 1 / 2
Therefore, the probability is 1/2 or 0.5.
I hope this explanation helps! Let me know if you have any further questions.
Numbers 1 - 24 that are divisible by 4: 4, 8, 12, 16, 20, 24
Which of those are also divisible by 3?