Probability Two Randomly Started Flower Bloom Periods Overlap

Two kinds of flowers will each begin blooming on a random day within the next $D$ days, independently of each other. Once it starts, the first kind stays in bloom for $d_1$ consecutive days and the second for $d_2$ consecutive days. What is the probability that at some moment both flowers are in bloom at the same time? Assume each flower's start day is uniform over days 1 through $D$, and a flower that starts on day $s$ is in bloom on days $s$ through $s + d - 1$, even if that runs past day $D$. Two flowers that are both in bloom on some common day count as overlapping. Derive an expression in $D$, $d_1$ and $d_2$, and explain how the answer changes if start times are continuous instead of whole days. The specific values are generated randomly. ```hint One number decides it Ask what single quantity built from the two start days determines whether the two bloom periods intersect. ``` ### Constraints and Clarifications - $D$, $d_1$ and $d_2$ are positive integers, and $d_1$ or $d_2$ may exceed $D$. - The two start days are independent, and each is uniform on $\{1, \dots, D\}$. ### Clarifying Questions - Is the start day a whole day, or a continuous time within the $D$-day window? - Must each bloom period finish within the $D$ days, or may it run past the end of the window? - Do two bloom periods that only touch at a boundary, one ending as the other begins, count as overlapping? ### What a Strong Answer Covers - Translating "both in bloom at some moment" into a condition on the two start days - Exact counting of the qualifying start-day pairs, including the asymmetric case where $d_1$ and $d_2$ differ - Bloom lengths that are as long as the window or longer - The discrete versus continuous difference and the off-by-one terms it creates ### Follow-up Questions - What is the expected number of days on which both flowers are in bloom? - If each bloom period must fit entirely inside the $D$-day window, how does the count change? - How would you extend the calculation to three flowers that must all be in bloom together?