Determining the number of times a rectangular object must be tossed.

Question

Suppose the four faces of a regular rectangular object are marked with letters A, B, C, and D, respectively. How can we determine the number of times this object must be tossed so that the probability of the ratio of numbers of D's to the number of tosses being between 0.15 and 0.35 is at least 0.95?

Answer 1

As stated, the problem seems incomplete. Please check your statement carefully. I will try to get you started on what I suppose the problem must be.

I assume you mean, that when a tetrahedral object is tossed, each of its four sides has an equal chance of facing downward. Thus the number $X$ of D's in $n$ independent tosses has $X \sim \mathsf{Binom}(n, 1/4),$ so that $E(X) = n/4$ and $Var(X) = 3n/16.$ Then you seek $n$ just large enough so that $P(0.15 < X/n < 0.35) > 0.95$

I suppose you are expected to standardize and use the normal approximations to these binomial distributions in order to solve for $n$.

Hint: Searching in R, I found the probabilities below.

diff(pbinom(c(.15,.35)*75, 75, .25))
[1] 0.9555702
diff(pbinom(c(.15,.35)*74, 74, .25))
[1] 0.9411401