Probability an integer chosen at random from 1 to 1000 is divisible by 3,5, or 7 A number is chosen at random from the first 1,000 positive integers. What is the probability that it's divisible by 3,5, or 7? 
So I started off by breaking the problem up and having:
divisible by 3: p(a)
divisible by 5: p(b)
divisible by 7 p(c)  
I know I'm going to apply the exclusion inclusion principle, but how do I find out how many numbers are divisible by each without going through all the numbers between 1 and 1000?

Use inclusion/exclusion principle:


*

*Include the amount of numbers divisible by $3$, which is $\Big\lfloor\frac{1000}{3}\Big\rfloor=333$

*Include the amount of numbers divisible by $5$, which is $\Big\lfloor\frac{1000}{5}\Big\rfloor=200$

*Include the amount of numbers divisible by $7$, which is $\Big\lfloor\frac{1000}{7}\Big\rfloor=142$

*Exclude the amount of numbers divisible by $3$ and $5$, which is $\Big\lfloor\frac{1000}{3\cdot5}\Big\rfloor=66$

*Exclude the amount of numbers divisible by $3$ and $7$, which is $\Big\lfloor\frac{1000}{3\cdot7}\Big\rfloor=47$

*Exclude the amount of numbers divisible by $5$ and $7$, which is $\Big\lfloor\frac{1000}{5\cdot7}\Big\rfloor=28$

*Include the amount of numbers divisible by $3$ and $5$ and $7$, which is $\Big\lfloor\frac{1000}{3\cdot5\cdot7}\Big\rfloor=9$


Hence the amount of numbers divisible by $3$ or $5$ or $7$ is:
$$333+200+142-66-47-28+9=543$$
