A bank vice-president declared that the probability that the bank’s funds were being embezzled was 0.70 and that the probability that an embezzlement would be discovered by the bank examiner was 0.90. Given that the bank examiner does not find the embezzlement and that the examiner would not discover embezzlement when none existed, what is the probability that the bank’s funds are not being embezzled?
To find the probability that the bank's funds are not being embezzled, we need to use Bayes' theorem.
Let's assign the following probabilities:
- P(A) = Probability of embezzlement (0.70)
- P(B) = Probability of not discovering embezzlement (0.10)
- P(A') = Probability of no embezzlement (1 - P(A) = 0.30)
We want to find P(A' | B), which is the probability of no embezzlement given that the embezzlement was not discovered.
Bayes' theorem states:
P(A' | B) = (P(B | A') * P(A')) / P(B)
In this case:
P(B | A') = 1, because the examiner would not discover embezzlement when none existed
P(A') = 0.30 (probability of no embezzlement)
To find P(B), we can use the law of total probability:
P(B) = P(B | A) * P(A) + P(B | A') * P(A')
Given that P(B | A') = 1, we don't need to calculate it. We only need to calculate P(B | A) and P(A).
To find P(B | A), notice that P(B | A) is the complement of the probability that the embezzlement would be discovered by the bank examiner, which is 0.90. Therefore:
P(B | A) = 1 - 0.90 = 0.10
Now we can calculate P(B):
P(B) = (0.10 * 0.70) + (1 * 0.30) = 0.07 + 0.30 = 0.37
Finally, we can calculate P(A' | B) using Bayes' theorem:
P(A' | B) = (P(B | A') * P(A')) / P(B) = (1 * 0.30) / 0.37 = 0.30 / 0.37 ≈ 0.81
Therefore, the probability that the bank's funds are not being embezzled, given that the embezzlement was not discovered, is approximately 0.81 or 81%.
To find the probability that the bank's funds are not being embezzled, we can use Bayes' theorem.
Let's define the events:
A = bank funds being embezzled
B = bank examiner does not find the embezzlement
We are given:
P(A) = 0.70 (probability of bank funds being embezzled)
P(B|A) = 0.90 (probability of bank examiner finding embezzlement given that it exists)
Bayes' theorem states that:
P(A|B) = (P(B|A) * P(A)) / P(B)
We need to find P(A|B), which is the probability of the bank's funds being embezzled given that the examiner does not find the embezzlement.
To calculate P(B), we can use the fact that the examiner would not discover embezzlement when none existed, so:
P(B) = 1 - P(A)
Substituting these values into Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
= (0.90 * 0.70) / (1 - 0.70)
= (0.63) / (0.30)
= 0.21
Therefore, the probability that the bank's funds are not being embezzled is 0.21, or 21%.