Question

Manufacturers are testing a new cologne before it's released. A machine uses sensors to determine if the cologne is working (maintaining a scent for 1 hour) 60% of the time. Another machine tests if the cologne works all the time (maintains a scent for over 5 hours). Find the probability that the cologne works all the time, given that it works 60% of the time.

Answer 1

To find the probability that the cologne works all the time given that it works 60% of the time, we can use Bayes' Theorem. However, since we don't have the value of P(B|A), we cannot calculate the exact value of P(A|B). We can make the assumption that P(B|A) is high and estimate that the probability of the cologne working all the time given that it works 60% of the time is 1.5 times the probability that the cologne works all the time.

Step-by-step explanation:

To find the probability that the cologne works all the time given that it works 60% of the time, we can use Bayes' Theorem. Let A be the event that the cologne works all the time and B be the event that the cologne works 60% of the time.

We are given that P(B) = 0.6 (the cologne works 60% of the time) and we need to find P(A|B) (the probability that the cologne works all the time given that it works 60% of the time).

Bayes' Theorem states:

P(A|B) = (P(B|A) * P(A)) / P(B)

Since we don't have the value of P(B|A) (the probability that the cologne works 60% of the time given that it works all the time), we cannot calculate the exact value of P(A|B).

However, we can make an assumption that P(B|A) is high, since the cologne working all the time implies that it would also work 60% of the time. Let's assume P(B|A) = 0.9 (90%).

Using this assumption, we can calculate an estimate for P(A|B):

P(A|B) = (P(B|A) * P(A)) / P(B) = (0.9 * P(A)) / 0.6 = 1.5 * P(A)

So, the probability that the cologne works all the time given that it works 60% of the time is 1.5 times the probability that the cologne works all the time.