Probability of missing the bus

Question

You arrive at bus stop some time uniformly distributed between $10:00$ and $10:15$ and bus leaves the bus stop sometime uniformly distributed between $10:00$ and $10:25$. What is the probability of you missing the bus?

Comments

yes :) i am getting same!

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thanks for the confirmation bro

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why you taking only half area in form of triangle?

Answer 1

Let X be a random variable denoting - number of minutes after 10:00 I arrive

Let Y be a random variable denoting - number of minutes after 10:00 the bus arrives

Probability density function of X = 1/15

Probability density function of Y = 1/25

Now for limits:

X can be any value b/w 0 and 15

but Y has to be b/w 0 and X for bus to leave me, also X and Y are indipendent -> the probability of bus leaving is not impacted by my arrival

$$
\int_0^{15} \int_0^{x} f(x) \cdot f(y) \, dy \, dx
$$

substituting uniform distribution constants for f(x) and f(y), we get

$$
\int_0^{15} \int_0^{x} \frac{1}{15} \cdot \frac{1}{25} \, dy \, dx
$$
 

Solving we get 0.3

Comments

Ignore this comment (Made accidentally)