# Probability of fails in a subject given data of a sample

I am leading with the following probability problem:

"The percentage of fails in a subject is 60%. If we know that in a sample of 20 pupils at least 8 of them have failed the exam, which is the probability of having less than 15?"

I have taken the probability of 0.6 and, given X="number of fails", thought of computing $$p(8\leq X\leq14)$$ in the sample using binomial distribution. In this way, for example: $$p(8)=\binom{20}{8}0.6^80.4^{12}$$ but the sum of all of them is $$0.8533$$, while the solution must be $$0.8717$$. I know I am applying something in a wrong way, but don't know what? Appreciate some help. Thanks.

• I think I have already found a way using conditional probability. The result equals to computing $$p(X<15|x\geq8)$$. Commented Feb 6, 2023 at 19:00

## 1 Answer

The problem of your attempt is that you failed to express the probability of interest correctly, which should be a conditional probability $$P[X < 15 | X \geq 8]$$, instead of $$P[8 \leq X \leq 14]$$ (as shown below, this unconditional probability is just the numerator of the answer), where $$X \sim B(20, 0.6)$$. Applying the conditional probability definition $$P(A | B) = P(A \cap B)/P(B)$$, it follows that
\begin{align} P[X < 15 | X \geq 8] = \frac{P[8 \leq X < 15]}{P[X \geq 8]}. \end{align} The best way to evaluate this probability is to use statistical software. For example, in R, $$P[8 \leq X < 15]$$ can be obtained by issuing sum(dbinom(8:14, 20, 0.6)) or pbinom(14, 20, 0.6) - pbinom(7, 20, 0.6), giving $$P[8 \leq X < 15] = 0.8533721$$. Similarly, $$P[X \geq 8] = 1 - P[X \leq 7]$$ can be obtained by running 1 - pbinom(7, 20, 0.6), giving $$P[X \geq 8] = 0.9789711$$, hence the final answer is \begin{align} P[X < 15 | X \geq 8] = \frac{0.8533721}{0.9789711} = 0.8717031. \end{align}