Surprise! As you may have already noticed, I am not stadarooni, but a new author to this blog. Rather than posting something that would correspond with the rest of the blog, I have chosen to write about something a bit different from the traditional post found on this blog. I will share with you how to identify the probability of something happening or in other words how many ways something can be done. I will try my best to not sound like a textbook!
The math behind it is quite simple, for example, let’s say we want to find out how many possible four digit codes we can make using numbers 0 to 9, we need to allocate four positions to assign with the number of possibilities to claim that position. So since it’s a four digit code we have four empty positions. For each position we assign the number of possibilities, in this case, there are ten possibilities since there is a total of ten digits. So in a mathematical form it would like this _ – _ – _ – _ (Each blank space represents a digit of the four digit code), and then we fill it with this 10 – 10 – 10 – 10 (Each blank is filled with the number of possibilities). Now we multiply each filled position, which in this case gives us 10,000. That means there are 10,000 possible four digit codes we can produce, or more specifically there is a 1 in 10,000 (0.01%) chance that we will get a certain code.
So now that we know how to calculate this, let’s calculate something a bit more interesting, say the probability of the forecast in fall. We know that it could be either sunny, cloudy, or rainy (assuming that whatever the weather is, it would remain the same all day). So we know we need to allocate seven positions for the week and fill each one with a number of possibilities, which in this case is three. So mathematically it would look like this 3 – 3 – 3 – 3 – 3 – 3 – 3 (Each spot is allocated with the number of possibilities). Then we do what we did before, which is multiply all of them together. We get a total of 2,187 possible weather patterns over the week (and thanks to technology, we can estimate which exact pattern it will be).
I find that extremely interesting, but there’s still something missing. What if we want to identify how many possibilities there are for let’s say the number of ways five people can stand in a line. We can’t use the same method as before because you can’t have the same person in more than one spot at a time, so we have to slightly change the math we use.
To do this, we start the same way. Since we are dealing with five people, we allocate five spots to be filled with each person and fill them with the number of possibilities. In the first position, we know it can be anyone, so it remains five. But in the next position we know that we will have one less person to choose from, because they are in the first position. This means that the number of possibilities reduces to four. So if we continue the same method, the math would look like this 5 – 4 – 3 – 2 – 1 (Each spot is allocated with how many people still need a spot in the line). We then multiply it together and get 120, meaning there are 120 different ways for five people to stand in a line.
Ok now let’s do something you yourself can calculate before we move on to something more challenging, like how many different combinations of clothes you can wear until you run out of new combinations. I will let you, the reader fill this one in as you go, and at the end, I will do my own just for the sake of example. So first you need to know how many clothes you have, and what type of clothing they are. You need to know the number of hats, shirts, sweaters, pants, shoes, and socks you have. If you’re interested in finding out, go count them now. If you happen to not have one of those types of clothing, just input one (since it will always be the same). Ok, now we know for the sake of this example that there are six types of clothing, so we allocate six spots. Mathematically this looks something like this _ – _ – _ – _ – _ – _ (Each empty spot represents a type of clothing). Now input the spots with the number of clothes you have, then multiply them together. For me the input looks something like this 3 – 9 – 4 – 5 – 2 – 12 which ends up being 12,960. That means assuming I wear one combination a day, I could go 12,960 days or 35 years until I come back to the same combination (and my wardrobe is pretty small!)
Alright, now that we know the uses of identifying probability, we come to the finale. Let’s calculate something that’s way more difficult. Let’s say we want to find out how many even six-digit numerals containing numbers 0 to 9 we can produce that have no repeating numbers in the numeral. We know the first number can’t be zero because that would be considered a five-digit numeral, the last digit has to be even and that digits cannot repeat throughout the six digit numeral. Mathematically it looks like this 8 – 8 – 7 – 6 – 5 – 4 (The last number has to be even, so there are four possible numbers, then we know the first number can’t be zero and since numbers can’t repeat and we’ve already assigned one of the numbers to the last spot, there are eight numbers we can still assign. Then the spots in-between the first and last we assign with numbers that still haven’t been used, and since we know 0 hasn’t been used on the first spot there will still be eight possibilities on the second spot). If we multiply these together we get a total of 53,760. As if that’s not already enough, we aren’t done!
Now we need to calculate what might happen if 0 were to be assigned to the last digit. That means that the first number would have nine possibilities because we know it can’t be zero since numbers can’t repeat. Everything else is the same, though. Mathematically it would look like this 9 – 8 – 7 – 6 – 5 – 1 (Since we are calculating just for the combinations that contain zero at the end, the last spot only has the one possibility. Due to that the first spot has nine possibilities since we cannot repeat any numbers). If we multiply these together we get a total of 15,120 combinations that will contain zero at the end. To finish this massive question off, we add both numbers together to finally get a grand total of 68,880 possible six-digit numerals!
Here are a few interesting facts that I calculated using this method.
- If a class has 30 kids and you line them up, there are 265 nonillions (30 zeroes) ways to line them up.
- There are an infinite number of different weather patterns over the course of a year since the number is so big it doesn’t have a name for it.
- Using average furniture per home, there are 3,628,800 ways to organize a room in your house, and 21,772,800 ways to organize your home.
Hopefully, I didn’t sound too much like a textbook, and that you learned something new or found it interesting. Please feel free to leave a comment with any interesting things you find using this method, or if you have a different method. Thanks for reading this abnormal blog post, and continue to enjoy stadarooni!