I would like to produce some kind of probability function/density for finding out the likelihood of a chosen number, $k$, to occur in a random sequence of length, $N$.
For Example:
Let's say we have a sequence of $N=1000$, random numbers in the range $\in [0,1]$. Furthermore, let's truncate the random numbers to have only 3 significant digits (0.001).
How can I calculate how many times a particular "even" modulo number is expected to occur, such that we have the probability of:
(a)$k \in [0.1, 0.2, …, 0.9]$(b)$k \in [0.01, 0.02, …, 0.09]$(c)$k \in [0.02, 0.05, 0.08]$
Possibly related questions:
- An integer is chosen at random from the first 1000 positive integers. Probability that is a multiple of both 6 and 8?
- Question about computing expected value of the limit of a geometric mean of random variables
PS. It's quite likely I have not stated the title of this question correctly, please feel free to adjust it.
Best Answer
Since you are looking only at numbers with three decimal places, your sample space has just $1000$ possible outcomes. For any particular set of $n$ outcomes the probability that each choice is in that set is just $p = n/1000$. (Your examples have $n= 9$, $9$ and $3$.) Since the choices are independent, the number of successes is a binomial distribution.
The expected number of successes in $N$ tries is then just $Np$.