Probability and compounding
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I came across an interesting probability problem yesterday. It was in Chinese, but I will translate it into English below.
The defensive missile can shoot down the offensive missile successfully 70% of the time. If the defense shoot 3 missiles toward the offensive missile, what’s the probability that they are successful? Think about it for a minute. My approach is shown below:
There is another way. Note in math and many other fields, it’s good to have a second approach for verification purpose. So the second approach goes like this: 0.7 + (1-0.7) x 0.7 + (1-0.7) x (1-0.7) x 0.7 = 0.7+0.21+0.063 = 0.973 = 97.3% (it matches the number above, hurray). Courtesy of my classmate in college (who is likely a math genius in high school).
Let me explain: 0.7 is the probability of hitting target the first time, (1-0.7)x0.7 is for the scenario of first time missing and second time hitting, vice versa for the last one (both first and second times missing) and last time hitting.
So there’s that. 97.3% is pretty high in terms of an outcome actually happening. What if the probability of hitting the target is 50% (roughly the odds of coin flipping, get head or tail). The answer is 1 – 0.5×0.5×0.5 = 87.5%. That’s still pretty decent odds. Extrapolate this to basketball, let’s say a player’s free throw hitting rate is 50%, he or she can be fairly certain (with 87.5%) that 1 point will be scored in 3 attempts. But note everything has a cost, hidden or not. Missile has a cost, free throw has a cost of earning it. I recall 15 years ago, when the NBA player Shaquille O’Neal was at his prime, he has a weakness of free throw. In his career he hits 52.7 percent from the free-throw line in his career (I googled it). And the opponents sometimes would use the “hack Shaq” strategy when he got the ball in the front court. Let’s calculate the probable points he would have made from two free throw attempts, note free throw is 1 point, so 0.527 + 0.527 = 1.054 is the possible score he can get from 2 free throws. I also looked at his career field goal percentage: 2 x 0.582 (2 pointers percentage, I am ignoring his 3 pointers for now, because he was not known for that) = 1.164, so in other words, it seems “hacking Shaq” strategy has some stats to back it for the opponents.
If anything, I would like to remind people lottery is a sure way to lose money if we know a bit probability. Use the popular Show Me Cash game as an example, the odds of winning are shown in the web page. Note the ticket cost 1 dollar. If any return below $1 means the player is losing. 1 x 9.6% + 10 x 1 / 102.6 + 250 x 1 / 3386.8 + 50,000* x 1 / 575,757 (the prize is at least 50,000 for this category, I would use 50,000 for simplicity) = 0.096 + 0.0975 + 0.0738 + 0.0868 = 0.3541 so let’s just say 35 cents for simplicity. Other games are similar, with possibility of even lower return (compared to 35% for Show Me Cash). Btw, every time I saw people use a stash of cash to buy lottery tickets at Schnucks or lottery machine, I am shaking my head internally. I understand sometimes people want to try luck, or believe their luck is better than the others, but in math this is not the way to get ahead in life. In my opinion the government should ban the lottery. Below is another tweet on the topic on probability and lottery n Chinese. Use google translate (with some editing from me): when we met with people last weekend and introduced each other, some people said that he had made software for casinos, and that no matter how good the gamblers were, you couldn’t play mathematics… When everyone did with eating, they started to talk about how to achieve financial freedom. My classmate who made casino software said that he is buying lottery tickets every day. I said, didn’t you say that a gambler couldn’t play math against casinos? He said he only buy a little… This tells me that no matter how educated and rational a person is, he can’t resist himself. The fluke mentality…(my comment: talk about psychology and discipline. One thing great about Warren Buffett is he would not bet on small things even if he knows it’s a good bet, again we are talking about discipline: He’s playing golf with a bunch of people and they all bet a dollar on a particular hole. Buffett won’t bet. “Why not, Warren?” they ask him, “you’re worth a gazillion.”He said, “I never break my discipline.”)
Compounding or compound effect
Sometimes called snowball effect. In investing there are a lot of talk on this. It’s applicable to other aspects of life as well, for example, exercise, eat / drink healthy, work, relationships and so on.
There is a reverse side of this effect too, for example, if a person does not eat healthy food, or have healthy life style, for example, drink a lot of sugar water, no exercise, he / she could gain weight pretty quickly too. And the detrimental effect to one’s body and mind will be growing fast as well. I will probably write a separate post on “compounding” in the near future.
PS: the story above on Buffett was a bit different from what I heard earlier. My version says someone offers Buffett a bet that Buffett will likely win. But his answer was similar: he said if I cannot resist the temptation of making $100 quick money, I cannot manage large sums of money.