Be careful - the part in green isn't true.
If the question were about the first 10 positive integers, then the probability would be 1/10, since only 1 of the first ten positive integers is divisible by 8 (that integer being 8)


The integers from 1 to 1000 inclusive have a nice feature:
1, 2, 3, 4, 5, 6, 7, 8, ___ 9, 10, 11, 12, 13, 14, 15, 16,___ 17, 18, 19, 20, 21, 22, 23, 24, ___ 25, 26, ...

As we can see, in every batch of 8 integers, exactly 1 is divisible by 8.
This patter continues all the way to 1000.
We have: ...___985, 986, 987, 988, 989, 990, 991, 992 ___ 993, 994, 995, 996, 997, 998, 999, 1000
So, all of our batches have exactly 8 integers.
As such, we can conclude that the probability is 1/8 that a selected number is divisible by 8

However, in the numbers from 1 to 10 inclusive, we have: 1, 2, 3, 4, 5, 6, 7, 8, ___ 9, 10
Here each batch does not have exactly 8 integers
So, we cannot conclude that the probability is 1/8 that a selected number is divisible by 8

Cheers,
Brent

125/1000

The best way to do this is to take all multiples of 8 till 125, as those will directly be divisible by 2 and uve got your answer.

Hello,

Small doubt,
I had entered the value 0.125 instead of 1/8. Would that be a problem ?

Thanks

the general rule to find number of terms is ((last term - first term)/difference between two consecutive terms) + 1 , so in our example the solution is ;
(1000-8)/8 + 1 = 124 + 1 = 125 terms
probability = 125/1000

how is it divided by 8

To me, this question is testing two things.
1) If you know how to find the lowest common multiple (LCM) and how to use it
2) If you know how to find the probability of an event


1) Finding LCM and using it

We are given 2 and 8, the LCM is 8.
Once you have found the LCM you can increment by that number to find all the common multiples for a series.
So for a series running from 1-1000, all of the common multiples of 2 and 8, will be multiples of 8
Divide 1000 by 8 and you get 125.
That's how many numbers in the series are multiples of 8, and therefore also multiples of both 2 and 8.


2) Finding probability of event

To find the probability we divide the number of outcomes that fulfill some given criteria by the total number of outcomes

P(2 and 8) = 125/1000

. we can divide both 125 and 1000 by 10, or in other words, move the decimal place left by one space for both and you get

P(2 and 8) = 12.5/100

anything divided by 100 is the same thing as "per cent" (literally per 100), so

P(2 and 8) = 12.5%

which is the same as 1/8. You can multiply 12.5 by 8 to see that this is true (the product is 100).


- Other approach
. As hamaireh73 pointed out you can use a formula for finding the number of values in an arithmetic series.
. if anyone would like a refresher on the derivation of that formula, and would like to see how it's applied here, I've attached a picture showing exactly that
. fair warning, some may find it tedious

Any number that is a multiple of 8 will also be a multiple of 2 as 8 = \(2^3\)
So, we need to find the number of integers divisible by 8 in the first 1000 positive integers = 125
Probability = \(\frac{125}{1000}\) = \(\frac{1}{8}\)

what about 4?

4 is a multiple of 2.

we need to find the number of integers divisible by 8 in the first 1000 positive integers = 125

Hope now is more clear.

Regards

I understand now. If it's not divisible by 8, then it won't be divisible by 2. We need numbers that are both divisible by 8 AND 2.

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