Interpreting Chance and Probabilities

mixed up cardsMuch of our lives are affected by random events; however, we are not fully aware of them. How could we, there is just too much randomness to keep track of. Despite our best efforts to reach our goals, we can’t eliminate the role of chance and unpredictability. We know that randomness exist, but to what extent? That is when a keen understanding of probabilities can be helpful. Calculating probabilities can determine a course of action and set realistic expectations for outcomes.

The insurance business is built on probabilities, which predict how often unforeseen events occur. That is, unforeseen for one particular individual, but almost a certainty for someone in a large group of people. For example: houses will burn down, cars will crash and people will die unexpectedly; we just don’t know who and where. Unfortunately, insurance cannot protect us from something bad happening (even the things we buy insurance for).

We like to think that we are in control of our lives. We tend to focus on intentions and give credence to willful actions or direct causes. Most of the time, when something works out for us we are eager to take credit. When something does not work out, we find fault by blaming ourselves or others. But I don’t think it’s that simple; success or failure is partly the result of chance (maybe as much as effort). Chance, however, does not mean that all outcomes have an equally chance of happening. Some outcomes are more probable than others, and sometimes it can be calculated.

How Do We Quantify Chance?

Although there are many unnoticed causes that we cannot quantify, there are situations when we can calculate chance. For example, a coin toss, the role of a die, and the dealing of playing cards. In simple situations, intuition is a reasonable guide. Simple mathematics instantly reveals the odds: There is a 1 in 2 chance of a coin landing on heads. There is a 1 in 6 chance of a die showing a six. There is a 1 in 52 chance of a turning over the ace of spade.

Problems arise when situations get more complicated. For instance, how many different possible outcomes are there for a 7 game series between 2 sports teams? From one team’s perspective, one outcome could be: win, win, loss, loss, win, loss and win. Like most probability questions, it can be calculated, but the answer is not immediately obvious. Assuming that all 7 games are played, there are 128 possible outcomes. In reality the outcomes are less, because after one team wins 4 games the series is over.

In fact, intuition is usually misleading. Why is that? 1) Humans are good at recognizing patterns, and often find patterns when there are none. 2) We tend to give more weight to recent events and stronger memories. 3) We are bias and notice what we look for. 4) Short-term results don’t always match actual probabilities, which will show up in larger sample sizes. The following are examples of how scrutinizing randomness can reveal surprising results.

Winning Super Bowls

Patriots teamAs a New England Patriots fan, I have enjoyed many exciting football games. For many years the Patriots were a perennial favorite to win the Super Bowl. From 2011 to 2015, they played in 5 consecutive AFC Championship Games (semi-finals). In those 5 years they won the Super Bowl once (the 2014 season). As an emotional fan, 1 Super Bowl victory in 5 semi-final appearances felt like a missed opportunity. They should have won more, I thought.

So was my initial reasoning sound? As it turns out: not really. I applied probabilistic thinking in 2 ways:

  1.  5 Consecutive AFC Championship Games (semi-finals): With 4 teams remaining at the end of the season, the chances are 1 in 4 that a chosen team will win the last two games. That’s assuming the 4 teams are equally talented, which is not always the case. But with a large sample size it should average out. In some years the Patriots were slightly better and in other years not as good. Nevertheless, the short-term result of 1 Super Bowl victory in 5 years is not surprising (the odds are 1 in 4).
  2. Team History: Then I considered the over-all team history. The Patriots have played in 12 AFC Championship Games in the Super Bowl Era. This includes a period, from 2001 to 2004, when the Patriots won 3 Super Bowls. The total numbers indicate that the Patriots have won 4 Super Bowls in 12 semi-final opportunities. That makes it 1 in 3, which beats the odds. So, what felt like an under achievement is actually a slight over achievement. Keep in mind that I am only calculating from the point of semi-final appearances. 5 consecutive AFC Championship games and 12 in total is way above average for a 32 team league.

The Monty Hall Problem

Let’s Make a Deal was a popular TV game show, which air in 1963 and ran for many years. The host was Monty Hall, and here is the problem: Monty Hall gave a contestant a choice, pick 1 of 3 doors. Behind one door was a valuable prize, and behind the other two was something far less valuable. Let’s say the contestant was playing to win a car. After the contestant picked a door (for example door #1), the host (who knew where the car was) opened one of the two remaining doors. Monty always opened a door with a dud prize (for example door #2).

Monty Hall ProblemThis is the point when the Monty Hall Problem arose. He gave the contestant the choice to change his/her mind. Should the contestant stick with door #1, or pick door #3. Without careful analysis, it seems that it makes no difference. Both door #1 and door #3 have an equal chance of winning the car. However, that is incorrect. The probability of winning is twice as high if the choice is switched.

The reasoning is very simple, yet it eludes many people. With the original choice, the odds are 1 in 3 that the car is behind the chosen door. That means that it’s 2 in 3 that the car is behind one of the other doors. When the host opens one of the dud doors (which he already knows has a dud prize), he is giving new information. He has eliminated one of the bad options. Therefore, there is a 2 in 3 chance of winning the car by switching doors (for example door #3), but only a 1 in 3 chance of winning by staying put.

We can exaggerate the game by imagining 100 doors. The contestant chooses 1 door and the host opens 98 doors without revealing the car. Remember that the host knows where the car is. The obvious choice here is to make the switch. There is only a 1 in 100 chance that the first choice is correct. That means that there is a 99 in 100 chance of winning the prize by switching doors.

Sharing a Birthday

birthday cakeIf you are hosting a party, what is the likelihood that two people will share the same birthday? Worded another way, how many people need to show up for the odds to be higher than 50%? Once again, intuition is shaky. One would think that the number would be quite high, as there are 365 days in a year. The answer is surprisingly low: just 23 guests will give a better than even chance of two people sharing a birthday.

The reason is that there are many possible combinations in which people can share a birthday. Each guest is not limited to matching a specific date on the calendar. Every arriving guest has the chance of matching a birthday with all the people already at the party. By the time it gets to 23 people, every guest has 22 chances of sharing a birthday with another guest.

My three examples above are fairly straight forward. Life is not as simple. Although we tend to feel responsible for the events in our lives, we should not underestimate the role of chance. Of all the possible outcomes, we don’t know how each day will turn out. We clearly can’t predict what will happen in life; however, there are isolated situations when information can help us determine the most probable outcomes. We need to figure out which facts apply and which facts do not apply. And unless we think it through, we can easily be fooled by surface information. Probabilistic thinking requires that at times we set our emotions and intuitions aside and let the numbers take over. Sometimes the numbers will reveal surprising results.

 

References: Leonard Mlodinow, The Drunkard’s Walk, (New York: Pantheon Books, 2008).


 

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s