SOUTH PLAINFIELD, NJ - When a trial or event which occurs over a finite period of time, it is one of the most misunderstood facets of basic probability. It is my hope that this column will do something to alleviate that misunderstanding.
PICK A NUMBER, ANY NUMBER
QUESTION #1: I ask you to pick any number which is greater than equal to one and less than or equal to ten, or more concisely – one to ten inclusive by rolling a 10-sided die. What is the probability that your number WILL NOT be picked?
Since I will only be successful 10% of the time, I will be unsuccessful 90% of the time – which also can be expressed as 9/10 or 0.9.
QUESTION #2: What is the probability that your number WILL NOT be selected if I roll the die ten times given the fact that the probability of being unsuccessful is 90%?
Because we are repeating the same trial with the same criteria, each trial is independent of one another, because when we do something such as roll a die, flip a coin, or spin a roulette wheel, or play a slot or video poker machine – any previous results do not have any impact on the present outcome or any future outcomes. Many of you might have heard the saying “the dice have no memory.” That is a superb non-numerical explanation which explains a numerically-based concept.
This time, our answer is a bit more complicated. You just take 0.9 and multiply it ten times (0.9^10), giving you answer of 34.87% - a bit higher than most people might think it would be.
What about repeating this 20 times or 30 times? The chances that your number will not come up when you repeat this 20 times, or two cycles is 12.16% - a little short of one out of every eight times. For 30 repetitions or three cycles, the probability of your number not coming up is 1.48% - or about once every 68 times; improbable but certainly not impossible.
WHAT ABOUT EXAMPLES WITH MORE NUMBERS TO CHOOSE FROM?
It’s the same thing, even with numerical computations that are a bit more difficult. A great example is that of a video poker machine. The probability of success when hitting a royal flush for most standard video poker games is 1/40,000. That means that the probability of NOT getting a royal flush in 40,000 hands is the number 0.999975 multiplied over and over 40,000 (0.999975^40,000) times – which amazing only reduces to probability of failure to 36.79%.
To keep things a bit more basic, we can just select any number between one and 40,000 inclusive. If we keep going and multiply 0.999975 over and over 80,000 times – we get 13.52%. Put another way, if you picked a number between one and 40,000 inclusive, there’s more than a one in seven chance that your number WILL NOT be picked.
Say you want to go to 120.000 or three cycles – there is 4.98% of that occurring, or a one in twenty chance that the number you select WILL NOT be picked.
QUESTION #3: Does this relate to lottery?
ABSOLUTELY, but analyzing that is much more complicated than it appears since the numbers that people are not all picked randomly (they do all have an equally likely chance of occurring). That will make a perfect article for another time.