# Conditional probability. What is the meaning of this explanation?

An insurance policy covers $2$ employees of Company Z. The policy will reimburse company Z for a maximum of $1$ loss per employee per year. It reimburses the full amount of the loss up to an annual company-wide maximum of $8000$. The probability of an employee incurring a loss in a year is $40$%. The probability that an employee incurs a loss is independent of the other employee's losses. The amount of each loss is uniformly distributed on $[1000, 5000]$. Given that one of the employees has incurred a loss in excess of $2000$, determine the probability that losses will exceed reimbursements.

So I'm not seeking a solution to this problem. I was able to solve it, but not for the reason stated in the solution. I don't understand what is meant by the following:

$[...]$ The probability that total losses exceed reimbursement $[...]$ is $\frac{1}{6}$ (this pat is not confusing). However, this is conditional on the occurrence of the loss of the employee whose loss is Y (but no longer conditional on the occurrence of the loss for the employee whose loss is $X$, because the event considered is already conditional on $X > 2000$, so the loss has occurred.) $[...]$

The full solution is rather wordy, so I restricted it to the portion that was confusing. (If more of it is needed in order to answer the post, then please let me know.)

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$[...]$ The probability that total losses exceed reimbursement $[...]$ is $\frac{1}{6}$.

This part shouldn't be confusing.

However, this is conditional on the occurrence of the loss of the employee whose loss is Y

This means that the total losses can exceed reimbursement only if the other employee (the one other than the one mentioned to have lost over $2000$) incurs a loss as well. This occurs with probability $.4$ (and is of course not a sufficient condition).

(but no longer conditional on the occurrence of the loss for the employee whose loss is $X$, because the event considered is already conditional on $X > 2000$, so the loss has occurred.) $[...]$

This means that while it is necessary that the employee mentioned incurs a loss, this is guaranteed as it is already mentioned that he has lost more than $2000$.

Basically, the point of the excerpt is to point out that in your working for the solution, the figure $.4$ should only appear once.

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In other words, the solution is $\frac{1}{6}\cdot0.4$. – Sarastro Feb 19 '13 at 2:38