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Question: Voice recognition software used by police detectives classifies voice recordings as either male or female. Each male voice recording is classified correctly with probability 0.95, while each female voice recording is classified correctly with probability 0.9. Suppose that each voice recording analyzed with the software originates from a man with probability 0.75 and from a woman with probability 0.25. Assume that all voice recordings analyzed are independent of each other.

(a) You are told that of the first 20 voice recordings analyzed, at least one was classified incorrectly. Given this information, calculate the probability that the first 5 voice recordings were all classified correctly.

For this question, I tried to do P(A|B), where P(B) is the probability of getting at least one incorrect result (using Poisson distribution) and P(A) is the probability of getting 5 voice recordings all classified correctly, which is (0.0625)^5.

However, I am not sure what P(A intersect B) is. I need that in order to calculate P(A|B).

I would be great if you could give me a hint.

Thanks.

Justin Chan
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  • Related, possibly useful: https://math.stackexchange.com/questions/2279851/applied-probability-bayes-theorem/2279888#2279888 – Ethan Bolker May 01 '18 at 21:50

1 Answers1

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The probability that a given recording is incorrect, given that at least one recording is incorrect, is $$\mathbb{P}(\text{Incorrect})=\dfrac{x}{20}$$

Where $x$ is the number of incorrect voice recordings. Analyzing the recordings in sequential order, the probability that the $n$th voice recording is incorrect is $$\mathbb{P}(n\text{th recording is incorrect})=\dfrac{x}{20-n}$$

Then the probability that the first $i$ recordings are correct is $$\mathbb{P}(n_1\cap n_2 \ldots \cap n_i \;\text{are correct}) = \prod_{n=1}^{i}1-\frac{x}{20-n}$$

From here, we can find the expected amount of incorrect voice recordings $\left(E(x)\right)$ and plug it in for $x$ $$E(x) = (0.75*20*(1-0.95))+(0.25*20*(1-0.9)) = 1.25$$

So then we plug this into our equation $$\prod_{n=1}^{5}1-\frac{1.25}{20-n} = 0.680616$$

Here's a graph of probabilities that the first $5$ are correct for different values of $x$

enter image description here

Joseph Eck
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