I'm planning on riding the bus to my friends house tomorrow (so i would like answers asap please) and i'm planning on taking my bike so we can ride together but im worried the bike rack on the bus is gonna be full and i have no idea how to handle that situation so if anyone could tell me what to do asap that would be fantastic.
Your responses to the suggestions so far show that you definitely need to take your bike on a bus, which is cool since it rules out all the other possibles variables that these:
So the bottom lines is how likely you are to be able to get your bike on a given bus. Let's call that number ρ.
Now then, since that probably is less than 1, you need to be prepared to wait for more than one bus. Let's call the number of busses you're prepared to wait for N.
Given those you can show the likelihood of NOT being able to get both you and your bike on one of the next N busses as:
( 1 - ρ ) ^ N
Here's the crux of your problem - this tells us that there's still a possibility you won't be able to get on the any of the first N busses, so you have to decide a reasonable confidence level for how much you need to make this journey. For the sake of argument, let me suggest a reasonable value of 95%. This gives us the formula
( 1 - ρ ) ^ N ≥ 0.05
1 - ρ ≥ N √ 0.05
ρ ≤ 1 - ( N √ 0.05 )
ρ is a local variable, which only you can know, so to express this in words:
If you think there's a 78% or higher chance of there being room for you and your bike on any given bus, be prepared to wait for 2 busses (i.e. try to get on the bus one before your latest possible bus)
If there's only a 63% - 78% chance of there being room for you and your bike on any given bus, be prepared to wait for 3 busses (i.e. try to get on the bus two before your latest possible bus)
If there's only a 53% - 63% chance of there being room for you and your bike on any given bus, be prepared to wait for 4 busses (i.e. try to get on the bus three before your latest possible bus)
And then you should have at least a 95% level of confidence of getting to your friend's on time.
You seem really anxious about this, which leads me to believe this has happened before. Here is what you can do: Take the front wheel, back wheel, and the pedals off your bike. Put all of the parts into a big box. Bike shops have boxes specially designed for this. Now you can take it into the bus itself and it won't matter whether the bike rack is full. Put it back together when you get to your friend's house.