6. October 2018

# What probability theory tells you about starting on time

Whether you are an event organizer waiting for participants to turn up or whether you are a bus driver waiting for passengers running to catch the bus, there is a dilemma: if you are so kind and wait for everyone to arrive, you make everyone else wait. Let’s explain the dilemma with probability theory.

Let’s say you are organising a group event with **n** participants. You advertise to all participants in advance that your event starts at a particular time **t**. Let’s assume that each participant arrives on time with probability **p**, where on time means before time **t**. Let’s also assume that participants arrive independently, i.e. that they don’t arrive in groups but all on their own schedule. Then the probability of all participants arriving on time is:

Let’s take an example with N=15 participants, where each participant is late in one out of twenty events on average. This means that the probability of all participants arriving on time is just about 46%.

What if each participant is late in one out of ten events? Then the probability of all participants arriving on time drops by more than a factor of two to less than 21%. This means that if you want to start the event with everybody present, you can do so only in one out of five events. Four out of five times, you will have to wait for one or more participants to arrive.

We can also ask different question: if you want the probability of all fifteen participants arriving on time to be greater than 90%, i.e. if you want to be able to start the event on time with all participants in nine out of ten events, then you find that every participant must be on time with probability more than 99.31% that is more than 99 out of 100 times!

It is a bit of a dilemma. On the one hand, you want to give everyone a chance to attend the event. On the other hand you do not want to let the participants wait who have turned up on time. To answer this question, we will leave probability theory aside.

One solution for you as an organizer could be to wait until time **t+m** in order to start the event a certain time margin **m** later than advertised. However, if you repeat that event, you might notice that your participants learn over time to sleep a little bit longer and try to arrive at **t+m** instead of the time you advertised. That’s not probability theory which says so, but simply my experience.

A second solution would be to wait until everyone arrives. However, now I predict that your participants will learn to try to arrive just before the last participant, because that means they aren’t late─only the last person to arrive can be considered late. Thus by conditioning the participants’ arrival time on the predicted time of the last participant, your advertised time has less influence on when participants actually show up.

A third solution is to start the event on time even if not all people who planned to attend have arrived yet. Under such circumstances you will start every event on time, assuming that you as an organizer are never late. This way, people learn over time that they have to show up on time in order to take part in the event. If you want to start events on time, this provides an overall positive feedback loop, although it risks a negative experience for those who arrive late.

Well, probability theory doesn’t give the answer. Probably.