Even the most dedicated mathematics student sitting in a lecture hall at one of the top US colleges inevitably has moments of wondering what it is all for. With areas like complex analysis and calculus, it is a reasonable question. However, statistical analysis is an area of math with real world applications that help it capture the imagination in a way that other areas can’t quite manage. Here, we will look at some real world practical applications of probability and expected value that add value both at work and play.
Using probability to calculate expected value
Most people are familiar with probabilities. Toss a coin and the probability of calling correctly is 0.5 or 50 percent. Roll a die and the probability of rolling a six is 1/6, or 0.167 or 16.7 percent.
With expected value, we can use probabilities to make informed decisions on the best course of action. Put simply, we multiply the probability of the outcome by its value, and that gives the overall expected value. It is best illustrated by way of example, so let’s take a look at some everyday applications in finance and gaming.
Choosing an upselling strategy
Suppose your business provides home insurance and you undecided between two potential upselling strategies that you could try with 500 existing customers:
- Adding on car insurance, which will net your business an additional $50 per sale on average
- Adding on life insurance, which will net your business an additional $70 on average
The market research team is confident that one in five existing customers will buy car insurance, while one in eight will buy life insurance. Which upselling strategy should you pursue?
Expected value of upselling car insurance is 0.2 x $50 x 500 = $5,000
Expected value of upselling life insurance is 0.125 x $70 x 500 = $4,375
This tells us that if all other things are equal, it will be more profitable to upsell car insurance. That proviso is an important one, however. There might be other factors to take into account, for example if selling car insurance take longer or involves more admin, that could affect your decision.
Selecting a strategy at the casino
Online casino gaming is a pastime that is getting more popular with every passing year. There are useful resources out there like GamblingSites.org to help you choose from the dozens of US casino sites that are available. But they can’t help when it comes to deciding how to place your wagers when you get inside. Perhaps expected value can help us make an informed decision.
Let’s use the game of roulette as an example, as it is one of the true icons of the casino. Suppose you are faced with an American roulette table and a European one. The former has the numbers one to 36 plus zero and double zero. There is no double zero on the European table. You decide that whichever table you play, you want to bet your dollar on red.
- European roulette: 18 pockets are red and 19 pockets are not. If the ball lands in red, you win a dollar (+1), if it does not, you lose a dollar (-1)
Expected value is (-1x 19/37) + (+1 x 18/37) = -2.7 percent
- American roulette: 18 pockets are red and 20 pockets are not due to the extra zero.
Expected value is (-1x 20/38) + (+1 x 18/38) = -5.26 percent
This tells us something we already knew, or at least strongly suspected, which is that a European roulette wheel is less harsh on our wallets, and has the lower house edge. When you think about it, that’s intuitively obvious, as a second zero pocket can be considered beneficial to the casino, and must increase the casino’s chance of winning. With the same payouts on offer on both tables, it clearly makes more sense to play European roulette.