This one required actually writing out the math, not just describing it.
Define random variables for bus arrival time and travel time, then express the event of making it before the movie starts as an inequality. Set up the probability as an integral over the joint distribution, and simplify using independence if applicable. Solve the integral, possibly by convolution or by conditioning on one variable.
Pro tip: Clearly state assumptions about independence and distribution shapes (e.g., uniform, exponential) before diving into integration; this shows you can translate a real-world scenario into a tractable probabilistic model.
Let X be the bus arrival time (relative to now) and T be the travel time to the cinema. Let M be the time until the movie starts. Assume X and T are independent random variables with known probability density functions f_X and f_T.
The person catches the bus at time X and arrives at the cinema at time X + T. They make it before the movie starts if X + T ≤ M. Thus, the probability is P(X + T ≤ M).
Express the probability as a double integral over the joint density: ∫∫_{x+t ≤ M} f_X(x) f_T(t) dx dt. If M is also random, include its distribution and integrate over it as well.
Condition on one variable, e.g., P(X + T ≤ M) = ∫ P(T ≤ M - x) f_X(x) dx. If M is fixed, this becomes ∫ F_T(M - x) f_X(x) dx, where F_T is the cumulative distribution function of T.
Plug in the specific distributions (e.g., uniform, exponential) and compute the integral analytically or numerically. State the final probability and discuss any assumptions or edge cases.
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