Many people recommend setting devices to grayscale to reduce screen time. The idea is that bright, vivid colors make apps like YouTube and Reddit more addictive. I tried it and found it helpful, but I kept having to turn colors back on to make sense of some websites. Eventually, I forgot to re-enable grayscale and relapsed into my old habits.
I came up with 2 solutions:
1. Set my screen to 5-10% saturation. This gives me nearly all the distraction-reducing benefits of grayscale while keeping just enough color to navigate UIs and interpret charts. I never have to toggle it off. This works great for PCs, Macs, and iPhone, but is impossible on Android. On PCs and Macs you may have productivity tools that use colors, so you can try desaturating your web browser only.
2. Having grayscale mode automatically re-enable after a fixed amount of time (say, 5 minutes). On Android, this can be done very easily with an app called Tasker, which costs $3.50.
The union bound states that P(⋃ni=1Ai)≤∑ni=1P(Ai) for any events A1,…,An. In practice, this is an enormously useful property. I think one reason it's so useful is that for rare events, it gives you essentially the same answer you'd get if the events were independent, but without proving anything about their dependence structure. To be precise, suppose A1,…,An are independent and all have probability p. Then, P(⋃ni=1Ai)=1−(1−p)n and ∑ni=1P(Ai)=np. Now note that 1−(1−p)n∈Θ(np) when np is bounded, so the union bound gives you asymptotically the same bound as assuming the events are independent.