When we roll a dice the probability of getting each number is (1/6). Apart from the fact that probabilities are supposed to be in the range of 0-1; why do they all add up to 1?
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The probability of something happening is 1. You just don’t know what it is yet. So the sum of all possible outcomes is 1
probabilities are measured “out of one” so to speak. the probability of *one* of the possibilities occuring is 1/1. the probability of rolling a 6 is (1/6)/1
“one out of one times, when you roll a die, you get a number between 1 and 6”
It represents the fact that *something* (as opposed to nothing) will happen with 100% probability.
They all add up to 1 because 1/1=100%. A probability is just x/1 or x/100% of something happening.
Because the results are mutually exclusive and comprehensive.
Mutually exclusive means that you can’t get more than one result. When you roll a die, it only shows one face. You can’t roll a 1 and a 2 simultaneously.
Comprehensive means it includes all possible outcomes. When you roll a die, it has to show one of the faces. It can’t show no face, or some other possibility.
If the possible outcomes were not mutually exclusive then the probabilities could add up to something greater than 1, because some outcomes could get double-counted.
If the possible outcomes aren’t comprehensive then the probabilities could add up to something less than 1, because some outcomes wouldn’t get counted at all.
One is the whole. 100%. It can’t be more or less. If you roll a standard 6 sided dice there is a 100% chance you’ll roll a number between 1 and 6.
Each probability is a number smaller than one. Let’s use a 20 sided dice because the math is easier for me. The chance of you rolling one of those numbers is 1/20. 1 divided by 20 equals 0.05. 0.05 = 5%. There is 20 possibilities, and you are dividing the number one by 20. 1 is all of the possible outcomes added together. In this instance 0.05 × 20 = 1.
Imagine you have a large sheet of paper 1 sq. meter in size. Then divide divide the paper into six rectangles of equal area. The first rectangle is marked as 1, the second is marked as 2, and so on up to rectangle 6. Then throw a dart at the paper at a random location.
Since each rectangle has an area of 1/6 sq. meters, the probability of hitting a particular rectangle is 16.̅6% (assuming your dart actually hit the paper). If you then add up the areas of all 6 rectangles, you’ll find they sum to 1 sq. meter.
It’s similar with probabilities. Die rolls: 1, 2, 3, 4, 5, and 6 are the only possibilities and it’s certain that only one of those possibilities will result from a roll (can’t roll two or more numbers from a single die). By definition, a certain event has a probability of 1. So the sum of the probabilities must add up to 1.
In the summer of 1976, on an oak living room table in my parents’ house, a die did in fact not land on a any of its sides. I was playing Yahtzee with my nephews, when one of the dice landed on one of its edges!! None of us have never seen this happen again. How unlikely is this? I don’t know, but I’ve always considered the probability of rolling either one of 1 to 6 is EXTREMELY close to 1, but not 1.
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