Finding the Probability of a Lost Marble Based on Conditional Probability

[pprex]

Homework Statement

There are 4 identical boxes filled to the top with marbles. Your friend is playing around with your favorite marble and accidentally drops it into one of the 4 boxes (assuming that all marbles look almost identical). Now it is equally likely that your marble could be in any of the 4 boxes. Using a sorter to try to find your lost marble, the probability of the machine finding your marble from any box, if it is in fact there, is 0.75. How after sorting through the first box, the machine does not turn up your marble. What is the probability that the marble is in box 1, 2, 3, 4?

Homework Equations

I understand this problem to be some sort of conditional probability. P(marble being in box i | machine could not find it in box 1).

The Attempt at a Solution

All I can get from this so far is conditional probability. I can see that if the marble was in fact in box 1, there is a 25% chance that the machine just couldn't find it because of its own error and that remains the same for all 4 boxes.

Please help.

 

[mighty2000]

Thank you for posting this problem. I would approach this problem using the concept of conditional probability. Let's define some variables first:

A: event that the marble is in box 1
B: event that the machine does not find the marble in box 1

We are given that P(B|A) = 0.25, which means that there is a 25% chance that the machine will not find the marble in box 1 even if it is there. We are also given that the machine has a 0.75 probability of finding the marble from any box if it is there. This means that P(B|not A) = 0.25.

Now, we can use Bayes' theorem to find the probability of the marble being in box 1 given that the machine did not find it in box 1:

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

P(A|B) = 0.25 * 0.25 / 0.25 = 0.25

This means that there is a 25% chance that the marble is in box 1 given that the machine did not find it in box 1. Similarly, the probability of the marble being in box 2, 3, or 4 would also be 0.25.

I hope this helps you solve the problem. Remember to always define your variables and use the appropriate formulas to solve problems like these. Good luck!