My everyday return commute from office to home while I lived in Bengaluru comprised the following three journeys: Two bus rides and an autorickshaw ride to cover the last 1 Km to my home. I rarely managed to get an autorickshaw ride so I usually used to try my luck and ask for a lift/ride from two-wheeler riders after alighting from the 2nd bus ride. I had the following rule: try your luck and continue asking for a ride for 15 mins. After having waited for 15 mins (and hence not being lucky to get a ride), just walk home for the last 1 Km.

Now here is the kicker: During my 1 year of stay in Bengaluru (around 230 workdays) I did NOT manage to get a lift for ONLY around 10 times. I got a lift for all the remaining days! My System 1 thinking concluded that people from Bengaluru are very kind and helpful!

Daniel Kahneman in his celebrated book “Thinking, Fast and Slow” defines two approaches that our brain uses to make inferences: System 1 and System 2.

  • System 1 is impulsive, intuitive, and extremely fast in taking decisions based on whatever we perceive. Examples of situations where System 1 is active: When you are browsing Twitter feeds or watching a news channel (quick to jump to conclusions) or when you sense danger and multiple other situations that demand a quick response.
  • System 2 on the other hand is capable of reasoning, very slow, methodical, and analytical in making an inference. Secondly System 2 is lazy: most of the time it simply endorses the interpretations of System 1! It is at work when you are solving an exam, trying to remember something, filing your taxes, comparing products, etc. A lot of effort and energy (glucose) is required while your System 2 is engaged.

Anyway, so without exerting any extra effort I quickly concluded based upon my experience that people from Bengaluru are very kind and helpful. Now let us take a step back and employ System 2 to understand if probability and uncertainty play a role here.

My objective was to get a lift within 15 minutes or walk home the last 1 Km. Let us define the following two events and for simplicity (we will deal with more real-life situations later on) let us assume that each of these is equally likely:

And let us say, that within 15 mins you make 3 attempts to ask for a ride. And in each attempt, you may succeed (1: get a lift and ride home) or fail (0: the rider ignores you). Thus the total number of possibilities (which we call the universe of possibilities) for 3 attempts is 2³=8.

Let us enumerate all the 8 possibilities (or journeys) and their corresponding probabilities:

In the above table A1, A2 and A3 are Attempt 1, Attempt 2, and Attempt 3 respectively. Now some important points:

  1. “0 0 1” means not getting a lift in the first two attempts and getting a lift in the 3rd attempt.
  2. Every attempt is independent, hence not getting a lift in an attempt does not influence the chance of outcomes in your next attempt.
  3. In every attempt, the probabilities of all the outcomes (in this case only two) should sum to 100%.
  4. The probability of every journey is computed by taking the product of the probabilities of every attempt. So probability of journey “0 0 1” = (Probability of NOT getting a ride in the 1st attempt) x (Probability of NOT getting a ride in the 2nd attempt) x (Probability of getting a ride in the 3rd attempt)
  5. The sum of the probabilities of all the journeys should be equal to 100%

It is no surprise that every journey in the above scenario is also equally likely since the individual events are equally likely. So how do we compute the probability of getting a ride? Two approaches:

  1. Sum the probabilities of journey number 2 (0 0 1) through journey number 8 (1 1 1) which comes out to be 87.5%
  2. Compute the probability of NOT getting a lift in any attempt (0 0 0) and subtract it from 1. I like this approach: 1 minus {probability of not getting a lift in any attempt.} which is also equal to 87.5%

Now, some astute readers might say, if I get a ride in the first attempt I’m done, why should I care about the remaining two attempts? Basically the last 4 journeys: 100, 101, 110, and 111. That’s correct. In this case, the probability is 50% (getting a lift on the very first attempt). The sum of these 4 journeys is also equal to 50%.

Hence there is yet another approach to compute the probability of getting a ride. You may end up with EITHER of the following scenarios:

  1. Manage to get a ride in the very 1st attempt (1: Probability = 50%)
  2. Fail in the first attempt but succeed in the 2nd attempt (“0 1”: Probability = 0.5²)
  3. Fail in the first two attempts but manage to get a ride in the 3rd attempt. (“0 0 1”: 0.5³)

The Sum of all these 3 probabilities is also equal to 87.5%

Now, what if, I decide to make 4 attempts to get a ride. Does it increase my chance of getting a lift? There are 16 possible journeys: 2⁴

Probability of getting a lift = 1 minus (Probability of never getting a lift) = 1–0.5⁴= 93.8%! Just by making one more attempt, I’m increasing the chance of getting a lift from 87.5% to 93.8%! Don’t confuse this with the coin toss experiment. You cannot increase the chance of getting ahead/tail just by tossing it several times. Here we are defining success as getting a lift in ANY of the attempts. Even in the coin toss experiment, if you define success as getting ahead in ANY of the attempts, you can maximize the chance of succeeding by tossing the coin several times.

What if I decide to make 10 attempts?

Probability of getting a ride = 1- 0.5¹⁰ = 99.9%!!

Every day during those 15 mins of waiting for a ride I usually made around 10 attempts. Now does this explain why I managed to get a ride in 220 days out of 230 days? Partly yes — but the reality is more complex than you think.

First of all, we have assumed that both the events: Getting a lift and not getting a lift are equally likely. This is not necessary. Consider the following:

Now the probability of never getting a lift (the very first journey) is 24%. Hence the probability of succeeding: getting a lift in any 4 attempts is 76%

Yet another possibility: we have implicitly assumed that in all 4 attempts the probabilities of either getting a lift or not getting a lift (whatever their values are) are the same. But this is also not necessary! The only condition is the probabilities of all possible events must sum to 100%. Consider the following:

Here the probability of getting a lift comes out to be 1–4.3% = 95.7%

Now what if we conduct this experiment during the starting days of the covid crisis when there was so much fear about the disease and people were worried about stepping outside their homes (forget about offering a ride to someone):

In this extreme case, the probability of never getting a ride is 96.1%. Hence the probability of getting a ride comes out to be equal to 3.9%!! Now you might say, this is an extreme case especially during the corona crisis. What if during the normal times a few of the riders had a bad day at work? Remember the probability of you getting a ride is decided by the rider who may think about various things before offering you a ride. Some riders might have a personal belief to never offer any ride to any stranger (probability of NOT getting a lift = 100%). Secondly, people behind in the road might simply follow the actions of the first rider whom you ask. If that rider ignores you, others behind may do the same.

Therefore, it is extremely hard to capture reality through a model, essentially because we just have no clue about the prior probabilities. But we can always be creative and think about the possible alternate scenarios which did NOT happen to get a better idea about reality.

Author: Ameet Chitnis

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