A microcontroller could do what you want. Think of a microprocessor as a device that makes other things work at set times and duration. E.g, when SW1 is operated make sound2 for 5 seconds and send it to speaker1 and flash LEDs 1-5 alternately for 6 seconds. It's not a magic box, just a very fast one.
At this stage asking for a circuit is way too vague. The pcb is relatively easy, the functionality (sounds, lights etc.) not so easy. Someone could design and build one but it probably wouldn't be what you want, so what then, build another?
it should cause one of two actions to occur at random when the trigger is pulled, preferably at a ratio of 5:6 where one outcome occurs slightly more often then the other
This may make perfect sense to you, but to me it means nothing. What sort of random action, why do you want a 5:6 ratio. In short what the hell are you trying to do? As I said before a Microprocessor isn't a magic box.
It might be worth your while to do some research into what you actually want. How is the trigger operated, and why? What type of switch do you want? what sort of noise?
Where is the user going to be relative to the LEDs. A small dimly lit LED won't IMO do much to influence behaviour. A flashing sequence of colours might attract my attention more.
What power source will be used? What sort of quality do you want from the speaker?
You don't need to pysically do everything to start with, e.g. Google 4cm speaker reviews (or similar) videos then you can hear the quality or otherwise of 4cm speakers. Let's say you now want a 10cm speaker because of its quality, only problem is it will be too big and a new problem, it needs too much power i.e. a big battery. Time to compromise.
Once you've figured out what you think you need, then start DIY. Arduino is very popular because it's cheap and there are lots of ready made modules (shields) and it's fairly easy to program. The Shields are designed for people who just want to make it go without the hassle of building every module and minimal programming.
You may say that you don't have time, but have you ever wondered why Degree Students working in Astro or other techie stuff use lathes and other machines to make things for their project? Asking a trained machinist to make a device to measure impact speed when landing on Mars is unlikely to produce the desired result. It's easier to train the student to learn how to use a lathe than train the machinist to understand the required physics or explain it with sufficient detail.
The are lots of ppl without degrees who can program a Microprocessor (an Arduino is a mini project board with a microP onboard).
I don't have a degree and I decided to learn how to program after breaking the stepper motor (2x) hand controller of an astro telescope mount. Like most things the 1st few steps were the hardest. Once I'd mastered that I was sufficiently interested to continue learning. It may come in handy as a useful extra skill.