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So I was teaching my younger brother Probability and in order to give him some intuition, I told him that events that occur for certain have a probability of $1$. So events such as one dying will have a probability of $1$. Later, I was trying to deduce this result from the definition of Probability. We say that $$P(\text{Event})=\frac{\text{Number of favorable Outcomes}}{\text{Total number of Outcomes}}$$ If we consider $S=\{\text{You Die},\text{You Live\}}$ as the set of all outcomes and the Event $\text{You Die}\in S$ to be a favorable outcome then $P(\text{You Die})=1/2.$ However we know that $P(\text{You Die})=1.$ Is there a way to show that this claim can be deduced from the definition of Probability?

Student
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  • I do not think that the example $P(You Die)$ is a good example as to what probability mathematically means. The fact that we all die is an unequivocal truth, not left to chance. Your younger brother may also not develop a liking for math as a result of this example. In terms of favorable outcomes and total outcomes, why not look at a numbercube (also called die, no pun intended)... It is a good question nonetheless – imranfat Oct 26 '16 at 19:48
  • If you know that "You die" is the only possible outcome, then "You live" isn't in $S$. – Bobson Dugnutt Oct 26 '16 at 20:17

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That definition of probability depends on the assumption that all the outcomes are equally likely.

That sounds like a circular definition, since it uses the phrase "equally likely". It really is circular, though useful when it makes sense: for cards and dice and coin tosses which you assume are fair.

Defining probability for one time events (like your life, or a presidential election) is much harder. Mathematicians and philosophers and statisticians work hard at the job.

We will all die with probability 1.

We don't all pay taxes with probability 1. (https://en.wikipedia.org/wiki/Death_and_taxes_(idiom))

Ethan Bolker
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