Often finding out the opposite of what you want informs you more than success. I worked really hard to find combinations that would not work. They informed me more than anything.
No repeats and no 0 in the first spot left about 4500 possibilities for a 4 digit number. If your guess was completely wrong (No Bulls. No Creots), you eliminated bout 40% of the possibilities!
We also learned in math that sometimes the easier way to figure out the probability for an event was to figure out the probability it will not happen and subtract that from 1.
Let’s look at the birthday problem.
How large a group of people would you need to have at your party for there to be a greater than 50% chance that at least 2 will have the same birthday?
What you do is start by figuring out what are the chances no one will have any common birthdays.
Say person 1 has 365 out of possible days to choose. For person 2 to not have one in common, they have 364 out of 365 possible days to choose from. Person 3 has 363 out of 365 possible days to choose from.
And so one. To figure the probability of NOT having any birthdays in common, we simply multiply the probability for each person.
At 23 people we get this:
When you calculate this to get a probability of NOT having any birthdays in common you get 0.49.
This means that the probability that two of them will have a birthday in common MUST be 0.51 (1-0.49)
So, essentially there is a greater than even chance that in any soccer game (counting the center ref) that two players will have a birthday in common.
I learned at an early age to look to what was wrong or the opposite of what you wanted. Often that was easier to calculate and told you so much more.
Image: Akash Kataruka