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Consider the sample space Ω={,,,} and the event space  F=2Ω. Which of the following partial functions  ℙ can be extended to  a probability function on P?

ℙ({})=0.01,ℙ({})=0.09,ℙ({})=0.09,ℙ({})=0.81,

ℙ({})=0.01,ℙ({})=0.15,ℙ({})=0.03,ℙ({})=0.81,

ℙ({,})=0.5,ℙ({,})=0.5,ℙ({})=0.4

ℙ({,,})=0.5,ℙ({,})=0.4,ℙ({})=0.4

What does it mean if event space is $2^{\text{sample space}}$? I really can't understand the problem; please can somebody explain what is the event space here?

amWhy
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Aditya Verma
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  • The "event space" is the set of all subsets of outcomes. We very specifically want our probability measure to satisfy some basic properties: $P(\emptyset)=0, P(\Omega) = 1,$ that if $A\subseteq B$ then $P(A)\leq P(B)$, that $P(A\cup B)\leq P(A)+P(B)$, and that if $A\cap B = \emptyset$ that $P(A\cup B) = P(A)+P(B)$. As to the content of your question, see if anything breaks here. For instance, with the last, you should have $P({HH,TT,HT,TH})=1$ but you should also have $P({HH,TT,HT,TH})\leq P({HH,TT,TH})+P({HT,TH}) = 0.5+0.4=0.9$... a contradiction. – JMoravitz May 29 '20 at 13:35
  • My explanation is here. – ryang Mar 27 '22 at 15:05

1 Answers1

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For any set $X$, $2^X$ denotes the set of all subsets of $X$. Now events are subsets of the sample space $\Omega$. Then $2^{\Omega}$ denotes the set of all possible subsets of the sample space or the set of all possible events.

ShBh
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  • But how is ℙ({})=0.01? it is 1/16 right? – Aditya Verma May 29 '20 at 13:34
  • Can you modify your answer to calculate any one of the probability? – Aditya Verma May 29 '20 at 13:35
  • @AdityaVerma "but how is $P({HH})=0.01$? it is 1/16 right?" There are many different probability measures we could apply to any given situation. The example probability measure happening in each of the different problems are just that... examples, and you are tasked with seeing if those examples are valid examples or if they contradict something. How is the probability of ${HH}$ in that example equal to $0.01$? Because it was allowed to be and they told us it was. – JMoravitz May 29 '20 at 13:38
  • Perhaps you are getting hung up over the fact that the elements of the sample space are $HH, HT, TH, TT$ and this reminds you of another example... that of flipping coins... and you are expecting the probability measures in the given examples to follow the same distribution and properties as the probability measures that real life coins do (in which case $Pr({HH})$ should have been $\frac{1}{4}$, not $\frac{1}{16}$). That is not what is going on in this problem. Perhaps you would prefer to rename the elements in the sample space instead as $a,b,c,d$ to get away from thinking of coins. – JMoravitz May 29 '20 at 13:42