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A consulting firm was hired to develop an Engineering project. Based on their previous experience, the direction of this office knows that the time (in months) needed to perform this type of task behaves according to a cumulative distribution function F(x)

The consulting firm made a commitment to the contracting company to deliver the project ready in m months.

Since the deadline has not been met, the office requested two more months. What is the probability that the new commitment will be fulfilled?

Solution: (Am I on the right path?)

$$ P(X < m + 2| X > m) = \frac{P(X < m + 2 \cap X > m)}{P(X > m)} $$

$$ P(X < m + 2| X > m) = \frac{P(m < X < m + 2)}{1 - P(X < m)} $$

$$ P(X < m + 2| X > m) = \frac{P(X < m + 2) - P(X < m)}{1 - P(X < m)} $$

$$ P(X < m + 2 | X > m) = \frac{F(m + 2) - F(m)}{1 - F(m)} $$

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    $\begingroup$ It looks like a reasonable approach but you ought to take care with the inequalities: half of them need to be non-strict inequalities. This makes a difference when $F$ has jumps (is not continuous) at either or both of $m$ and $m+2.$ $\endgroup$ – whuber Dec 29 '18 at 23:23
  • $\begingroup$ I defer to whuber always, but so long as the variables are continuous this looks the right way to head. If you have F(x), do the results look reasonable? $\endgroup$ – eSurfsnake Dec 30 '18 at 6:37

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