You have finally selected a washing machine which is gonna wash all your blues (no pun) and socks without complaining, for the rest of your life. Color- check, price- in budget, better than your ‘always boasting’ neighbor- check. Just about you are leaving the aisle and heading towards billing section with your happy machine; the overly-positive salesman makes you an offer you can’t refuse.
He asks you; whether you would be willing to buy an extended warranty for another 10K- which includes 1 year of extra cover after your company warranty ends. Now, as a smart buyer as you are- you are in a bit of dilemma; to take the extended warranty or not.
Probability to the rescue! A beer company actually ran a 2 Million $ experiment to help you take your extended warranty decision- more on this later. Let me first lay down some penny stocks about the math of practical- probability (its short and it’s not that boring).
Probability is everywhere. Boarding a bus; visiting a grocery store for your favorite snack; chances of your win in Ludo, winning a lottery, your insurance claim approvals and much more. It’s just that we don’t really care enough to work on the math behind it. Probabilities do not tell us what will happen for sure, they tell us; what is likely to happen and what is less likely to happen. It can help you with something as spectacular as winning a lottery, or something miserable such as investing in a down-falling stock. It’s all about numbers; it’s all about chances- but probability triumphs in the end.
Let’s do some brain-storming. When it comes to risk, our fears always gets the better of us by making us conveniently ignore the actual risk in figures. According to a striking finding from Freakonomics-by Steve Levitt and Stephen Dubner – swimming pools in the backyard of houses are far dangerous than guns in the closet. A child under the age of 10 is 100 times more likely to die in a swimming pool than from a gun accident.
Another paper published in Cornell found that thousands of Americans may have died since the September 11 attacks because they were afraid to fly! When more Americans opted to drive rather than to fly after 9/11, there were estimated 240+ additional traffic deaths per month or the following 3 months in 2001. Obviously, this fear has faded as the fear of global terrorism. diminished. As ridiculous as it may sound- apparently now people are afraid to fly because of Covid. Sadly standing true to this- we all have seen plights of poor labors who died in hundreds while taking the road towards their home. Long story short- the law of large numbers confirms that if more people take the road- accidents are going to increase too. The law states that as the number of trials increases, the average of the outcomes will get closer and closer to its expected value- more light on this later.
The law also explains why casinos always win over a longer duration of game and is also your easiest answer to why you shouldn’t buy the extended warranty on your machine.
Let’s help you buy your machine first. Pondering over probability theory in a billing section queue isn’t the best way to spend weekend- I agree!
The entire insurance company is built on probability and statistics. When you buy an insurance; in exchange of a regular payment, you have transferred certain risks of having your machine broken. Basically, if your machine breaks down within a year post brand-warranty, the insurance company pays for the repair/ exchange machine in worst case scenarios. But, have you thought for a moment, why these companies are willing to take those high damage risks for a small amount? The answer is; they are gonna make a fortune of money from hundreds of insurance payers – while they are going to compensate only handful of customers for their broken machines. Let me back myself up with some stats. Pointer number one- the insurance company selects products which are hardly broken- they don’t insure your local SP road products. They choose the bigger brands which has a good track record of product run-life. And to do that all they need is a ‘normal’ looking Statistician (no pun). They basically find out the percentage of broken machines over a hundred machines — and choose the least broken machines to play their bet on. If they take 10K as extended warranty from those 100 customers and replace machines of -lets say 5 customers- they are taking home ‘make me rich’ kind of money home.
In probability world, ‘Expected value’ is a magical spell. It’s this word that tells you that playing roulette is a loss-making business even in your best of days.
Grab your coffee, we are of to understand all of this by an example. Let’s say I buy a lottery ticket worth one rupee. And the company issues one lakh tickets for a certain lucky draw of say- INR 50,000. My chances of winning is 1/100000- which is lesser than me getting hit by an meteorite. Now, my expected value of winning amount can be calculated by taking the weighted sum of various outcomes by their respective probabilities. For example- if I buy 1 ticket, my expected winning amount would be- 1 * ( 1/100000), which is tantamount to 0.000001.
Similarly, if I buy a thousand tickets, my expected winning amount is- 1* 0.000001 + 2 * 0.0000001 + 3* 0.000001 – – – 1000* 0.000001 – which is somewhere close to 0.499- probably not the best bet still. No, please don’t worry- you don’t have to spend your weekend calculating this manually. We can write a python/Julia code and find this out- something like-
Interesting thing to note here- I will have to buy 1500 tickets to ensure that my earning is at least 1 rupee. You read that right. That way, if I have to ensure my winning of the jackpot amount, I would have to buy ten lakh tickets. Hmm, investing INR 10 Lakhs to get INR 50,000- I don’t call myself statistician but this isn’t the best of a deal, is it?
This is a clear callout- ‘Don’t try to hack probabilities- at least not the casino/lottery ones’. The casino house/lottery company is always on the better side of the pond. You might be feeling all the luck in the world inside you, but the chances of your win for any instance is an independent event and is never affected by the previous outcomes. This is called ‘Gambler’s Fallacy’.
A good example of which probably we can relate well is a game of Ludo- everybody has got a 6, and you just can’t seem to be getting it- if you think after 15 throws of die your chances of getting a 6 increases- then honestly I don’t have delighting answers for you. The die of a Ludo has no memory; previous throws of the die will not influence next throws. It can’t remember that you have not got a 6 for about 15 throws. Your chances of getting 6 on every throw is still 1/6. You can keep cursing your fate till the cows come home but its totally out-of-your-control thing. This right here is called a Bernoulli trial. You either get success or you get a failure (6 or no 6). The Bernoulli distribution is the discrete probability distribution of a random variable( throw of die here) which takes a binary values: 1 with probability p, and 0 with probability (1-p)- where 1 is getting a 6, 0 is not getting a 6.
The Binomial distribution describes the behavior- the outcome of n random experiments, each having a Bernoulli distribution with probability p. More on this here- Bernoulli distribution, Binomial distribution. The Binomial equation can be calculated as:
Long story short, insurance companies are interested in finding out the binomial distribution of broken washing machine and then make profits on pricing their premium right. So, you know what to say to the sales gentleman when asked about the infamous ‘extended warranty’ trap! Congratulations on your new washing machine!
Stay safe! Thanks for reading.