A number is randomly chosen in interval 1 to 3. what is prob that first digit to right of decimal is 5
My attempt:
1.0, 1.1 ....3.0 --> total there are 21 digits
favourable outcomes = 1.5, 2.5 --> means there are 2 favourable outcomes.
so required probability = 2/21 = 0.095
But in textbook, he assumed it as uniform distribution and solved it(then the ans = .1). When to apply which distribution, how to interpret? Please elaborate
Since we only care about the number immediately to the right of the decimal place, we can just look at the range $[0,1)$ instead of $[1,3)$. (The choice of including the endpoints is kind of irrelevant since it makes a difference of 1 number when there are infinite numbers in any range. I picked it the way I did so the process for finding the answer is simpler).
Consider the following number in that range:
$$0.2748191\dotsm$$
Notice how we can make a similar looking number that does fit the requirements:
$$0.5748191\dotsm$$
In fact, for any combination of digits after the first digit after the decimal point, you can make 10 numbers by changing the number after the decimal point.
$$0.0748191\dotsm$$ $$0.1748191\dotsm$$ $$0.2748191\dotsm$$ $$0.3748191\dotsm$$ $$0.4748191\dotsm$$ $$0.5748191\dotsm$$ $$0.6748191\dotsm$$ $$0.7748191\dotsm$$ $$0.8748191\dotsm$$ $$0.9748191\dotsm$$
You’ll see that only 1 out of 10 possibilities satisfy the requirement of having a 5 immediately after the decimal point. Therefore, you get the answer $0.1$