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  1. I think that the probabability that Obams wins the next elections is 80%.
  2. I am 80% sure that Obama wins the next elections.

Are those two statements equivalent?

Denoting event $A$ as "Obama wins the next elections", I would interpret statement 1. as

  1. $P(A) = 0.8$

and statement 2. as

  1. $P(P(A) = 1) = 0.8$

suggesting they are different. However, I feel that in daily usage, those two statements are used very interchangeably, hence my confusion.

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  • $\begingroup$ If in reality $P(A)=0.8$ then $P(A)=1$ is plainly false. I don't think you could say that P(false)=0.8 because false is not a random event or the like. $\endgroup$
    – Richard Hardy
    Commented Jan 15, 2015 at 11:28
  • $\begingroup$ Well but we don't know the true value. Furthermore, I'm not even sure whether this is the correct way of interpreting the two statements, hence my question. $\endgroup$
    – bonifaz
    Commented Jan 15, 2015 at 11:39
  • $\begingroup$ Yeah, I don't know these things too well, too. But another note: I don't think I am 80% sure that Obama wins the next elections. is equivalent to $P(P(A)=1)=0.8$. To me both statement 1 and statement 2 look more like $P(A)=0.8$. $\endgroup$
    – Richard Hardy
    Commented Jan 15, 2015 at 11:44

1 Answer 1

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Avoiding to go into the mathematical complexities that one could go into here, the expression

$P(A)$ essentially is a shortcut for $P(\{A\; {\rm is\; realized\}})$. Moreover $P(A) = 1$ is equivalent ("almost surely") to $\{A\; {\rm is\; realized\}}$.

So

$$P(P(A) = 1) = P(\{A\; {\rm is\; realized\}}) = P(A)$$

Of course, unending discussions and explorations can arise if one wants to discuss issues of objective/subjective probability, measure theory etc.

ADDENDUM
Responding to a comment by the OP, the verbal statements surrounding the $0.8$ number are both too vague in order for anything respectably "definite" to be said about them or their relation. Admittedly, they do sound different: one could argue that the first one is closer to an expression of an estimated objective probability, while the second, more personalized and categorical in tone, leans more towards an expression of a subjective degree of belief.

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  • $\begingroup$ This states that at least the two mathematical statements in the questions are equivalent. But are also the two original statements equivalent (i.e., is my interpretation of the narrative statement into mathematical statements correct) ? $\endgroup$
    – bonifaz
    Commented Jan 15, 2015 at 13:14
  • $\begingroup$ I think you ought to go into the "mathematical complexities" because I cannot (at this time) see any possible interpretation of what you wrote that is mathematically correct and consistent with the question. If you're going to write something like "$P(P(A)=1)$", then you really need to explain how you are assigning a probability distribution to $P(A)$. $\endgroup$
    – whuber ♦
    Commented Jan 15, 2015 at 17:47
  • $\begingroup$ @whuber But, I don't do that. Following exactly what the OP writes, I assign a probability to the event $\{P(A) =1\}$ which is equivalent, a.s. I guess, to the event "A is realized". Note that in my answer I did not comment on whether I agree or not with how the OP translated into symbols the verbal expressions, nor whether these expressions represent correct/consistent use of mathematical concepts, symbols and relations. $\endgroup$
    – Alecos Papadopoulos
    Commented Jan 15, 2015 at 21:17
  • $\begingroup$ "$\{P(A)=1\}$" makes no sense as an "event" except perhaps as a shorthand for a set of the form $\{A\,|\,P(A)=1\}$. Notice that "$A$" does not enter here; it is a bound variable in the sense that "$\{P(B)=1\}$" is an equivalent description. As such, this expression cannot possibly say anything about a specific set $A$. I think you might be running into trouble by attempting to use expressions that make no mathematical sense as if they were meaningful. You are not required to accept nonsense statements that appear in a question; you should feel free to correct them. $\endgroup$
    – whuber ♦
    Commented Jan 15, 2015 at 21:52

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