# Exponential Distribution - conditional Probability

I have the following question as practice for an upcoming test:

A website suggests that men think about sex an average of $15$ times per day.

Let $x$= the number of hours that pass between successive thoughts of sex, = $1.6$

At least $1.6$ hours have passed since the last thought of sex. What is the probability that in total, at least $3.2$ hours will pass until the next thought of sex?

I thought since Exponential Distributions are memoryless, that this would just be $P(x \ge 3.2)$, but apparently this is wrong.

• The exponential distribution would be a terrible fit for this problem, but I assume that's just given to you in the question? – Steven Irrgang Mar 14 '18 at 4:56
• What makes you say it's a terrible fit? The waiting time of a poisson random variable is an exponential random variable. Are you referring to the fact that it depends on what time of day? – Remy Mar 14 '18 at 4:59
• Calling it a "poisson random variable" is basically just restating the same flakey assumption (for the reason you gave). But yes, time of day, one event triggering another, common cause of multiple events, pretty much every way you could imagine it failing. Take even just this specific question, if 1.6 hours have passed there's a fair chance he's asleep, and so the actual probability is a lot higher than the answer for this distribution. – Steven Irrgang Mar 14 '18 at 5:37

You're right that the exponential distribution is memoryless. That is,

$$P(X\geq 3.2 \mid X \geq 1.6)=P(X\geq1.6)$$

Perhaps the wording of the question threw you off.

It's looking for the probability that you have to wait at least $3.2$ hours in total so $1.6$ more hours.

which of course comes out to be

\begin{align*} P(X\geq1.6) &=1-P(X\lt1.6)\\\\ &=1-\int_0^{1.6} \frac{1}{1.6}e^{-\frac{x}{1.6}}dx\\\\ &\approx 0.36788 \end{align*}

• ahhhhhhhhhhhh Yes you're right. Thank you. – MattyS11 Mar 14 '18 at 5:12
• I read it incorrectly originally too. Easy mistake! – Remy Mar 14 '18 at 5:13