# Probability that the proportion of a shorter segment with relation to the longer one is less than $\dfrac{1}{4}$

The problem is as follows.

We randomly pick a point on a segment line of lenght L.

What is the probability that the quotient of the shorter segment with relation to the longer one is less than $\dfrac{1}{4}?$

The first was to say that let be X a random variable such that for every point p $$X(p)=\dfrac{p}{L-p}.$$ But there is no reason to say that any point has more probability to be chosen that just another different point, so we can say that X is a uniformly distributed random variable.

So we have $$p_X\left(\dfrac{1}{4}\right)=P\left(X<\dfrac{1}{4}\right)=\int_0^{\frac{1}{4}}\dfrac{1}{L}dt=\dfrac{1}{4L}.$$ BUT I feel that this is not the correct approach, so I made a second attempt.

SECOND ATTEMPT.

We want that $$\dfrac{p}{L-p}<\dfrac{1}{4}$$ so with some algebra we get that $$p<\dfrac{L}{5}.$$

Now we do $$\int_0^{\frac{L}{5}}\dfrac{1}{L}=\dfrac{1}{5}.$$

But also this doesnt satisfies me.

Can you give me a hand/hint please?

• Wouldn't it just be $\frac{2}{5}$? Unless im missing something or read your question wrong. To get a quotient of $\frac{1}{4}$ between the big line segment and the small one, the dot must be placed in the first 1/5th of the line, or the last 1/5th. That would make so the short line is at least 1/4 the size of the big line. So... 2/5? – Loocid Apr 17 '15 at 4:32
• @Loocid: That is a good answer. Please post it that way. – Ross Millikan Apr 17 '15 at 4:35
• But formally how can I argue that? – HeMan Apr 17 '15 at 4:35
• You show, as Loocid has, if the break is in one region the ratio is less than $1/4$ and if it is in the rest the ratio is greater. Then you compute the length of the first region. – Ross Millikan Apr 17 '15 at 4:39

Maybe thinking in terms of percentages or decimals will make more intuitive. If the length is $L$ and the point is at $p$ and $p$ is less than $L-p$, then we want $\frac{p}{L-p}<.25$ or $p<0.2L$. And if $p>L-p$ then we want $\frac{L-p}{p}<.25$ or $p>0.8L$
If p is uniformly distributed across L, the probability of it being less than $0.2L$ or greater than $0.8L$ is $0.4$ or $\frac{2}{5}$ as given in the comments above.