Compute bus-on-time and dice-sum probabilities

You are asked two basic probability questions.

A) Catching a bus to arrive before a movie

A person is going to the cinema. The movie starts in L minutes.

• Buses depart the stop exactly every I minutes at times: $0, I, 2I, \dots$ .
• The bus ride time from the stop to the cinema is a constant R minutes.
• The person arrives at the bus stop at a random time that is uniformly distributed over one bus interval, i.e. $T \sim \text{Uniform}(0, I)$ , where $T$ is the time since the last bus departed.

The waiting time is therefore $W = I - T$ (with $W \in (0, I]$).

Task: Derive $\Pr[W + R \le L]$, i.e. the probability the person arrives at the cinema before the movie starts, as a function of $I, R, L$.

(Optionally: evaluate it for a concrete example such as $I=10$, $R=25$, $L=40$.)

B) Sum of two dice

Two fair six-sided dice are rolled.

Task: For an integer $S$ in $[2,12]$, compute $\Pr[d_1 + d_2 = S]$.

Clarify how the probability changes with $S$ (e.g., give a formula or a small table).