CHM 421/621 Statistical Mechanics

Lecture 5

Introduction and ReviewLecture Plan

Review of probability theory • Definitions of simple and compound events

• Frequency and probability

• A priori and a posteriori probabilities

• Disjoint and non-disjoint distributions

• Conditional Probabilities and Independent events

Nature of the distributionhOi = 1

Q

Zdr

Zdp O(r,p)exp (��H(r,p))

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

⇠Z

dE Ō(E) g(E) exp(��E)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

The peak and width of the distribution are decided by the temperature and density of states of the system.

E 0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

f(x)g(x)

f(x)*g(x)exp(��E)

AAAB/HicbVBNS8NAEN3Ur1q/oj16WSxCPVgSEfRYFMFjBfsBTSib7bRdutmE3Y0YQv0rXjwo4tUf4s1/47bNQVsfDDzem2FmXhBzprTjfFuFldW19Y3iZmlre2d3z94/aKkokRSaNOKR7AREAWcCmpppDp1YAgkDDu1gfD312w8gFYvEvU5j8EMyFGzAKNFG6tnlzJMhhsd4Uj31AtAE35z07IpTc2bAy8TNSQXlaPTsL68f0SQEoSknSnVdJ9Z+RqRmlMOk5CUKYkLHZAhdQwUJQfnZ7PgJPjZKHw8iaUpoPFN/T2QkVCoNA9MZEj1Si95U/M/rJnpw6WdMxIkGQeeLBgnHOsLTJHCfSaCap4YQKpm5FdMRkYRqk1fJhOAuvrxMWmc116m5d+eV+lUeRxEdoiNURS66QHV0ixqoiShK0TN6RW/Wk/VivVsf89aClc+U0R9Ynz9UfJPkAAAB/HicbVBNS8NAEN3Ur1q/oj16WSxCPVgSEfRYFMFjBfsBTSib7bRdutmE3Y0YQv0rXjwo4tUf4s1/47bNQVsfDDzem2FmXhBzprTjfFuFldW19Y3iZmlre2d3z94/aKkokRSaNOKR7AREAWcCmpppDp1YAgkDDu1gfD312w8gFYvEvU5j8EMyFGzAKNFG6tnlzJMhhsd4Uj31AtAE35z07IpTc2bAy8TNSQXlaPTsL68f0SQEoSknSnVdJ9Z+RqRmlMOk5CUKYkLHZAhdQwUJQfnZ7PgJPjZKHw8iaUpoPFN/T2QkVCoNA9MZEj1Si95U/M/rJnpw6WdMxIkGQeeLBgnHOsLTJHCfSaCap4YQKpm5FdMRkYRqk1fJhOAuvrxMWmc116m5d+eV+lUeRxEdoiNURS66QHV0ixqoiShK0TN6RW/Wk/VivVsf89aClc+U0R9Ynz9UfJPkAAAB/HicbVBNS8NAEN3Ur1q/oj16WSxCPVgSEfRYFMFjBfsBTSib7bRdutmE3Y0YQv0rXjwo4tUf4s1/47bNQVsfDDzem2FmXhBzprTjfFuFldW19Y3iZmlre2d3z94/aKkokRSaNOKR7AREAWcCmpppDp1YAgkDDu1gfD312w8gFYvEvU5j8EMyFGzAKNFG6tnlzJMhhsd4Uj31AtAE35z07IpTc2bAy8TNSQXlaPTsL68f0SQEoSknSnVdJ9Z+RqRmlMOk5CUKYkLHZAhdQwUJQfnZ7PgJPjZKHw8iaUpoPFN/T2QkVCoNA9MZEj1Si95U/M/rJnpw6WdMxIkGQeeLBgnHOsLTJHCfSaCap4YQKpm5FdMRkYRqk1fJhOAuvrxMWmc116m5d+eV+lUeRxEdoiNURS66QHV0ixqoiShK0TN6RW/Wk/VivVsf89aClc+U0R9Ynz9UfJPkAAAB/HicbVBNS8NAEN3Ur1q/oj16WSxCPVgSEfRYFMFjBfsBTSib7bRdutmE3Y0YQv0rXjwo4tUf4s1/47bNQVsfDDzem2FmXhBzprTjfFuFldW19Y3iZmlre2d3z94/aKkokRSaNOKR7AREAWcCmpppDp1YAgkDDu1gfD312w8gFYvEvU5j8EMyFGzAKNFG6tnlzJMhhsd4Uj31AtAE35z07IpTc2bAy8TNSQXlaPTsL68f0SQEoSknSnVdJ9Z+RqRmlMOk5CUKYkLHZAhdQwUJQfnZ7PgJPjZKHw8iaUpoPFN/T2QkVCoNA9MZEj1Si95U/M/rJnpw6WdMxIkGQeeLBgnHOsLTJHCfSaCap4YQKpm5FdMRkYRqk1fJhOAuvrxMWmc116m5d+eV+lUeRxEdoiNURS66QHV0ixqoiShK0TN6RW/Wk/VivVsf89aClc+U0R9Ynz9UfJPk

g(E) exp(��E)AAACAXicbVDLSgNBEJyNrxhfq14EL4NBSA6GXRH0GJSAxwjmAdklzE56kyGzD2ZmxbAkF3/FiwdFvPoX3vwbJ8keNLGgoajqprvLizmTyrK+jdzK6tr6Rn6zsLW9s7tn7h80ZZQICg0a8Ui0PSKBsxAaiikO7VgACTwOLW94M/VbDyAki8J7NYrBDUg/ZD6jRGmpax71S7XyJHVEgOExHpfOHA8UwbVy1yxaFWsGvEzsjBRRhnrX/HJ6EU0CCBXlRMqObcXKTYlQjHIYF5xEQkzokPSho2lIApBuOvtgjE+10sN+JHSFCs/U3xMpCaQcBZ7uDIgayEVvKv7ndRLlX7kpC+NEQUjni/yEYxXhaRy4xwRQxUeaECqYvhXTARGEKh1aQYdgL768TJrnFduq2HcXxep1FkceHaMTVEI2ukRVdIvqqIEomqBn9IrejCfjxXg3PuatOSObOUR/YHz+AGvUlZE=AAACAXicbVDLSgNBEJyNrxhfq14EL4NBSA6GXRH0GJSAxwjmAdklzE56kyGzD2ZmxbAkF3/FiwdFvPoX3vwbJ8keNLGgoajqprvLizmTyrK+jdzK6tr6Rn6zsLW9s7tn7h80ZZQICg0a8Ui0PSKBsxAaiikO7VgACTwOLW94M/VbDyAki8J7NYrBDUg/ZD6jRGmpax71S7XyJHVEgOExHpfOHA8UwbVy1yxaFWsGvEzsjBRRhnrX/HJ6EU0CCBXlRMqObcXKTYlQjHIYF5xEQkzokPSho2lIApBuOvtgjE+10sN+JHSFCs/U3xMpCaQcBZ7uDIgayEVvKv7ndRLlX7kpC+NEQUjni/yEYxXhaRy4xwRQxUeaECqYvhXTARGEKh1aQYdgL768TJrnFduq2HcXxep1FkceHaMTVEI2ukRVdIvqqIEomqBn9IrejCfjxXg3PuatOSObOUR/YHz+AGvUlZE=AAACAXicbVDLSgNBEJyNrxhfq14EL4NBSA6GXRH0GJSAxwjmAdklzE56kyGzD2ZmxbAkF3/FiwdFvPoX3vwbJ8keNLGgoajqprvLizmTyrK+jdzK6tr6Rn6zsLW9s7tn7h80ZZQICg0a8Ui0PSKBsxAaiikO7VgACTwOLW94M/VbDyAki8J7NYrBDUg/ZD6jRGmpax71S7XyJHVEgOExHpfOHA8UwbVy1yxaFWsGvEzsjBRRhnrX/HJ6EU0CCBXlRMqObcXKTYlQjHIYF5xEQkzokPSho2lIApBuOvtgjE+10sN+JHSFCs/U3xMpCaQcBZ7uDIgayEVvKv7ndRLlX7kpC+NEQUjni/yEYxXhaRy4xwRQxUeaECqYvhXTARGEKh1aQYdgL768TJrnFduq2HcXxep1FkceHaMTVEI2ukRVdIvqqIEomqBn9IrejCfjxXg3PuatOSObOUR/YHz+AGvUlZE=AAACAXicbVDLSgNBEJyNrxhfq14EL4NBSA6GXRH0GJSAxwjmAdklzE56kyGzD2ZmxbAkF3/FiwdFvPoX3vwbJ8keNLGgoajqprvLizmTyrK+jdzK6tr6Rn6zsLW9s7tn7h80ZZQICg0a8Ui0PSKBsxAaiikO7VgACTwOLW94M/VbDyAki8J7NYrBDUg/ZD6jRGmpax71S7XyJHVEgOExHpfOHA8UwbVy1yxaFWsGvEzsjBRRhnrX/HJ6EU0CCBXlRMqObcXKTYlQjHIYF5xEQkzokPSho2lIApBuOvtgjE+10sN+JHSFCs/U3xMpCaQcBZ7uDIgayEVvKv7ndRLlX7kpC+NEQUjni/yEYxXhaRy4xwRQxUeaECqYvhXTARGEKh1aQYdgL768TJrnFduq2HcXxep1FkceHaMTVEI2ukRVdIvqqIEomqBn9IrejCfjxXg3PuatOSObOUR/YHz+AGvUlZE=

g(E)AAAB63icbVBNS8NAEJ3Ur1q/qh69LBahXkoigh6LInisYD+gDWWz3aRLdzdhdyOU0L/gxYMiXv1D3vw3btoctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHR2nitA2iXmsegHWlDNJ24YZTnuJolgEnHaDyW3ud5+o0iyWj2aaUF/gSLKQEWxyKarfnQ+rNbfhzoFWiVeQGhRoDatfg1FMUkGlIRxr3ffcxPgZVoYRTmeVQappgskER7RvqcSCaj+b3zpDZ1YZoTBWtqRBc/X3RIaF1lMR2E6BzVgve7n4n9dPTXjtZ0wmqaGSLBaFKUcmRvnjaMQUJYZPLcFEMXsrImOsMDE2nooNwVt+eZV0Lhqe2/AeLmvNmyKOMpzAKdTBgytowj20oA0ExvAMr/DmCOfFeXc+Fq0lp5g5hj9wPn8AIGeNnw==AAAB63icbVBNS8NAEJ3Ur1q/qh69LBahXkoigh6LInisYD+gDWWz3aRLdzdhdyOU0L/gxYMiXv1D3vw3btoctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHR2nitA2iXmsegHWlDNJ24YZTnuJolgEnHaDyW3ud5+o0iyWj2aaUF/gSLKQEWxyKarfnQ+rNbfhzoFWiVeQGhRoDatfg1FMUkGlIRxr3ffcxPgZVoYRTmeVQappgskER7RvqcSCaj+b3zpDZ1YZoTBWtqRBc/X3RIaF1lMR2E6BzVgve7n4n9dPTXjtZ0wmqaGSLBaFKUcmRvnjaMQUJYZPLcFEMXsrImOsMDE2nooNwVt+eZV0Lhqe2/AeLmvNmyKOMpzAKdTBgytowj20oA0ExvAMr/DmCOfFeXc+Fq0lp5g5hj9wPn8AIGeNnw==AAAB63icbVBNS8NAEJ3Ur1q/qh69LBahXkoigh6LInisYD+gDWWz3aRLdzdhdyOU0L/gxYMiXv1D3vw3btoctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHR2nitA2iXmsegHWlDNJ24YZTnuJolgEnHaDyW3ud5+o0iyWj2aaUF/gSLKQEWxyKarfnQ+rNbfhzoFWiVeQGhRoDatfg1FMUkGlIRxr3ffcxPgZVoYRTmeVQappgskER7RvqcSCaj+b3zpDZ1YZoTBWtqRBc/X3RIaF1lMR2E6BzVgve7n4n9dPTXjtZ0wmqaGSLBaFKUcmRvnjaMQUJYZPLcFEMXsrImOsMDE2nooNwVt+eZV0Lhqe2/AeLmvNmyKOMpzAKdTBgytowj20oA0ExvAMr/DmCOfFeXc+Fq0lp5g5hj9wPn8AIGeNnw==AAAB63icbVBNS8NAEJ3Ur1q/qh69LBahXkoigh6LInisYD+gDWWz3aRLdzdhdyOU0L/gxYMiXv1D3vw3btoctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHR2nitA2iXmsegHWlDNJ24YZTnuJolgEnHaDyW3ud5+o0iyWj2aaUF/gSLKQEWxyKarfnQ+rNbfhzoFWiVeQGhRoDatfg1FMUkGlIRxr3ffcxPgZVoYRTmeVQappgskER7RvqcSCaj+b3zpDZ1YZoTBWtqRBc/X3RIaF1lMR2E6BzVgve7n4n9dPTXjtZ0wmqaGSLBaFKUcmRvnjaMQUJYZPLcFEMXsrImOsMDE2nooNwVt+eZV0Lhqe2/AeLmvNmyKOMpzAKdTBgytowj20oA0ExvAMr/DmCOfFeXc+Fq0lp5g5hj9wPn8AIGeNnw==

Density of states

Averaged over all {r,p}

s.t. H({r,p}) = E

⇠ E 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

Definitions: Simple EventSimple Event: Distinct or mutually exclusive results of an experiment.

E.g. Tossing a coin: 2 simple events possible (H/T)

A single die is rolled: 6 simple events possible

Frequency: The ratio of the number of times a simple event occurs to the total number of times the experiment is performed.

E.g. A coin is tossed 10 times and the result H occurs 4 times. Then the frequency of occurrence of H in the experiment is fH = 2/5

The frequency of a particular event might depend on the number of times the experiment is conducted.

It is, however, desirable to associate a characteristic number with the event that does not depend on the number of trials.

Definitions: ProbabilityProbability of a simple event:

Two possible interpretations:

(1) Carrying out N successive expts. on the same system.

(2) Carrying out 1 expt. each on N identically prepared (macroscopically

indistinguishable) systems. => Ensemble

We shall assume that either approach would yield the same value for the probability of the event. => Ergodic Hyopthesis

Pi = limN!1

fi(N) = limN!1

ni(N)

NAAACLXicbVDLSgMxFM3UV62vqks3wSLUTZkRQTdCUReuSgX7gE4ZMmmmDc1khuSOUIb5ITf+igguKuLW3zB9LLTtgcDhnHO5ucePBddg22Mrt7a+sbmV3y7s7O7tHxQPj5o6ShRlDRqJSLV9opngkjWAg2DtWDES+oK1/OHdxG89M6V5JJ9gFLNuSPqSB5wSMJJXvK97HN9gV/DQS2suRC6XAYwyHHi8XDs31grPDRShqZwmsrSWecWSXbGnwMvEmZMSmqPuFd/dXkSTkEmggmjdcewYuilRwKlgWcFNNIsJHZI+6xgqSch0N51em+Ezo/RwECnzJOCp+nciJaHWo9A3yZDAQC96E3GV10kguO6mXMYJMElni4JEYIjwpDrc44pRECNDCFXc/BXTATFVgCm4YEpwFk9eJs2LimNXnMfLUvV2XkcenaBTVEYOukJV9IDqqIEoekFvaIw+rVfrw/qyvmfRnDWfOUb/YP38Am+Zp60=AAACLXicbVDLSgMxFM3UV62vqks3wSLUTZkRQTdCUReuSgX7gE4ZMmmmDc1khuSOUIb5ITf+igguKuLW3zB9LLTtgcDhnHO5ucePBddg22Mrt7a+sbmV3y7s7O7tHxQPj5o6ShRlDRqJSLV9opngkjWAg2DtWDES+oK1/OHdxG89M6V5JJ9gFLNuSPqSB5wSMJJXvK97HN9gV/DQS2suRC6XAYwyHHi8XDs31grPDRShqZwmsrSWecWSXbGnwMvEmZMSmqPuFd/dXkSTkEmggmjdcewYuilRwKlgWcFNNIsJHZI+6xgqSch0N51em+Ezo/RwECnzJOCp+nciJaHWo9A3yZDAQC96E3GV10kguO6mXMYJMElni4JEYIjwpDrc44pRECNDCFXc/BXTATFVgCm4YEpwFk9eJs2LimNXnMfLUvV2XkcenaBTVEYOukJV9IDqqIEoekFvaIw+rVfrw/qyvmfRnDWfOUb/YP38Am+Zp60=AAACLXicbVDLSgMxFM3UV62vqks3wSLUTZkRQTdCUReuSgX7gE4ZMmmmDc1khuSOUIb5ITf+igguKuLW3zB9LLTtgcDhnHO5ucePBddg22Mrt7a+sbmV3y7s7O7tHxQPj5o6ShRlDRqJSLV9opngkjWAg2DtWDES+oK1/OHdxG89M6V5JJ9gFLNuSPqSB5wSMJJXvK97HN9gV/DQS2suRC6XAYwyHHi8XDs31grPDRShqZwmsrSWecWSXbGnwMvEmZMSmqPuFd/dXkSTkEmggmjdcewYuilRwKlgWcFNNIsJHZI+6xgqSch0N51em+Ezo/RwECnzJOCp+nciJaHWo9A3yZDAQC96E3GV10kguO6mXMYJMElni4JEYIjwpDrc44pRECNDCFXc/BXTATFVgCm4YEpwFk9eJs2LimNXnMfLUvV2XkcenaBTVEYOukJV9IDqqIEoekFvaIw+rVfrw/qyvmfRnDWfOUb/YP38Am+Zp60=AAACLXicbVDLSgMxFM3UV62vqks3wSLUTZkRQTdCUReuSgX7gE4ZMmmmDc1khuSOUIb5ITf+igguKuLW3zB9LLTtgcDhnHO5ucePBddg22Mrt7a+sbmV3y7s7O7tHxQPj5o6ShRlDRqJSLV9opngkjWAg2DtWDES+oK1/OHdxG89M6V5JJ9gFLNuSPqSB5wSMJJXvK97HN9gV/DQS2suRC6XAYwyHHi8XDs31grPDRShqZwmsrSWecWSXbGnwMvEmZMSmqPuFd/dXkSTkEmggmjdcewYuilRwKlgWcFNNIsJHZI+6xgqSch0N51em+Ezo/RwECnzJOCp+nciJaHWo9A3yZDAQC96E3GV10kguO6mXMYJMElni4JEYIjwpDrc44pRECNDCFXc/BXTATFVgCm4YEpwFk9eJs2LimNXnMfLUvV2XkcenaBTVEYOukJV9IDqqIEoekFvaIw+rVfrw/qyvmfRnDWfOUb/YP38Am+Zp60=

Properties of ProbabilityInherent properties of probability:

If there is no apparent reason for one simple event to occur more frequently than any other then we can assume that their probabilities are all equal.

In such a case, if there are totally M events possible, then the above properties imply that

e.g. Throwing a single die: numbers occur with probability 1/6

Tossing an unbiased coin: H/T occur with probability 1/2

Picking any particular card from a deck of 52 cards: ?

(1) For any event i: Pi � 0AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPbUDbbSbt0s0l3N0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hodua3nlBpHssHM0nQj+hA8pAzaqz0WO9x0h3gmLi9csWtunOQVeLlpAI56r3yV7cfszRCaZigWnc8NzF+RpXhTOC01E01JpSN6AA7lkoaofaz+cVTcmaVPgljZUsaMld/T2Q00noSBbYzomaol72Z+J/XSU147WdcJqlByRaLwlQQE5PZ+6TPFTIjJpZQpri9lbAhVZQZG1LJhuAtv7xKmhdVz61695eV2k0eRxFO4BTOwYMrqMEd1KEBDCQ8wyu8Odp5cd6dj0VrwclnjuEPnM8fSgOP/w==AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPbUDbbSbt0s0l3N0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hodua3nlBpHssHM0nQj+hA8pAzaqz0WO9x0h3gmLi9csWtunOQVeLlpAI56r3yV7cfszRCaZigWnc8NzF+RpXhTOC01E01JpSN6AA7lkoaofaz+cVTcmaVPgljZUsaMld/T2Q00noSBbYzomaol72Z+J/XSU147WdcJqlByRaLwlQQE5PZ+6TPFTIjJpZQpri9lbAhVZQZG1LJhuAtv7xKmhdVz61695eV2k0eRxFO4BTOwYMrqMEd1KEBDCQ8wyu8Odp5cd6dj0VrwclnjuEPnM8fSgOP/w==AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPbUDbbSbt0s0l3N0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hodua3nlBpHssHM0nQj+hA8pAzaqz0WO9x0h3gmLi9csWtunOQVeLlpAI56r3yV7cfszRCaZigWnc8NzF+RpXhTOC01E01JpSN6AA7lkoaofaz+cVTcmaVPgljZUsaMld/T2Q00noSBbYzomaol72Z+J/XSU147WdcJqlByRaLwlQQE5PZ+6TPFTIjJpZQpri9lbAhVZQZG1LJhuAtv7xKmhdVz61695eV2k0eRxFO4BTOwYMrqMEd1KEBDCQ8wyu8Odp5cd6dj0VrwclnjuEPnM8fSgOP/w==AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPbUDbbSbt0s0l3N0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hodua3nlBpHssHM0nQj+hA8pAzaqz0WO9x0h3gmLi9csWtunOQVeLlpAI56r3yV7cfszRCaZigWnc8NzF+RpXhTOC01E01JpSN6AA7lkoaofaz+cVTcmaVPgljZUsaMld/T2Q00noSBbYzomaol72Z+J/XSU147WdcJqlByRaLwlQQE5PZ+6TPFTIjJpZQpri9lbAhVZQZG1LJhuAtv7xKmhdVz61695eV2k0eRxFO4BTOwYMrqMEd1KEBDCQ8wyu8Odp5cd6dj0VrwclnjuEPnM8fSgOP/w==

(2) Normalisation:X

i2allPi = 1

AAACDnicbVA9SwNBEN3zM8avU0ubxRCwCnciaCMEbSwjmA/IHcfeZpMs2d07dueEcOQX2PhXbCwUsbW289+4Sa7QxAcDj/dmmJkXp4Ib8LxvZ2V1bX1js7RV3t7Z3dt3Dw5bJsk0ZU2aiER3YmKY4Io1gYNgnVQzImPB2vHoZuq3H5g2PFH3ME5ZKMlA8T6nBKwUudXAZDIQXHIwUc5xwBXOAy0xEWIywY2I4yvsR27Fq3kz4GXiF6SCCjQi9yvoJTSTTAEVxJiu76UQ5kQDp4JNykFmWEroiAxY11JFJDNhPntngqtW6eF+om0pwDP190ROpDFjGdtOSWBoFr2p+J/XzaB/GeZcpRkwReeL+pnAkOBpNrjHNaMgxpYQqrm9FdMh0YSCTbBsQ/AXX14mrbOa79X8u/NK/bqIo4SO0Qk6RT66QHV0ixqoiSh6RM/oFb05T86L8+58zFtXnGLmCP2B8/kD/3CbYQ==AAACDnicbVA9SwNBEN3zM8avU0ubxRCwCnciaCMEbSwjmA/IHcfeZpMs2d07dueEcOQX2PhXbCwUsbW289+4Sa7QxAcDj/dmmJkXp4Ib8LxvZ2V1bX1js7RV3t7Z3dt3Dw5bJsk0ZU2aiER3YmKY4Io1gYNgnVQzImPB2vHoZuq3H5g2PFH3ME5ZKMlA8T6nBKwUudXAZDIQXHIwUc5xwBXOAy0xEWIywY2I4yvsR27Fq3kz4GXiF6SCCjQi9yvoJTSTTAEVxJiu76UQ5kQDp4JNykFmWEroiAxY11JFJDNhPntngqtW6eF+om0pwDP190ROpDFjGdtOSWBoFr2p+J/XzaB/GeZcpRkwReeL+pnAkOBpNrjHNaMgxpYQqrm9FdMh0YSCTbBsQ/AXX14mrbOa79X8u/NK/bqIo4SO0Qk6RT66QHV0ixqoiSh6RM/oFb05T86L8+58zFtXnGLmCP2B8/kD/3CbYQ==AAACDnicbVA9SwNBEN3zM8avU0ubxRCwCnciaCMEbSwjmA/IHcfeZpMs2d07dueEcOQX2PhXbCwUsbW289+4Sa7QxAcDj/dmmJkXp4Ib8LxvZ2V1bX1js7RV3t7Z3dt3Dw5bJsk0ZU2aiER3YmKY4Io1gYNgnVQzImPB2vHoZuq3H5g2PFH3ME5ZKMlA8T6nBKwUudXAZDIQXHIwUc5xwBXOAy0xEWIywY2I4yvsR27Fq3kz4GXiF6SCCjQi9yvoJTSTTAEVxJiu76UQ5kQDp4JNykFmWEroiAxY11JFJDNhPntngqtW6eF+om0pwDP190ROpDFjGdtOSWBoFr2p+J/XzaB/GeZcpRkwReeL+pnAkOBpNrjHNaMgxpYQqrm9FdMh0YSCTbBsQ/AXX14mrbOa79X8u/NK/bqIo4SO0Qk6RT66QHV0ixqoiSh6RM/oFb05T86L8+58zFtXnGLmCP2B8/kD/3CbYQ==AAACDnicbVA9SwNBEN3zM8avU0ubxRCwCnciaCMEbSwjmA/IHcfeZpMs2d07dueEcOQX2PhXbCwUsbW289+4Sa7QxAcDj/dmmJkXp4Ib8LxvZ2V1bX1js7RV3t7Z3dt3Dw5bJsk0ZU2aiER3YmKY4Io1gYNgnVQzImPB2vHoZuq3H5g2PFH3ME5ZKMlA8T6nBKwUudXAZDIQXHIwUc5xwBXOAy0xEWIywY2I4yvsR27Fq3kz4GXiF6SCCjQi9yvoJTSTTAEVxJiu76UQ5kQDp4JNykFmWEroiAxY11JFJDNhPntngqtW6eF+om0pwDP190ROpDFjGdtOSWBoFr2p+J/XzaB/GeZcpRkwReeL+pnAkOBpNrjHNaMgxpYQqrm9FdMh0YSCTbBsQ/AXX14mrbOa79X8u/NK/bqIo4SO0Qk6RT66QHV0ixqoiSh6RM/oFb05T86L8+58zFtXnGLmCP2B8/kD/3CbYQ==

Pi =1

M8 i 2 {1,M}

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

A priori probabilitiesIf we have no information about the system then it is only fair to assume that all simple events are equally likely.

Probabilities calculated from such assumptions (without measurements) are termed as a priori probabilities.

Probabilities obtained from actual measurements are termed as a posteriori probabilities

When we set about calculating properties of macroscopic systems we will only know for certain the values of the macroscopic constraints (e.g. p,V,T,N) we impose on them. We would have no reason to favour less any microstate over others as long as they all conform to the constraints imposed.

Thus, in statistical mechanics we shall make the assumption of equal a priori probabilities for micro states consistent with the macroscopic constraints.

Definitions: Compound Events

Simple events can be thought of as points in a “sample space”.

Simple events Compound event

Compound Events: Any specified collection of points in sample space.

If we denote a compound event by A, then probability of occurrence of A is

P (A) :=X

i 2 APi

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

Compound Events: Example

What is the probability that a card drawn from a deck of 52 cards will have a

on it?

Solution:

Each card in the deck has the same a priori probability 1/52

There are 13 cards with on them.

Therefore,P ( ) =

X

i 2Pi

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

= 13⇥ 152

=1

4AAACD3icbVDLSgMxFM34rPU16tJNsCiuyqRWdFMounFZwT6gM5RMmmlDM5khyQhlmD9w46+4caGIW7fu/BvTdkBtPRA495x7ubnHjzlT2nG+rKXlldW19cJGcXNre2fX3ttvqSiRhDZJxCPZ8bGinAna1Exz2oklxaHPadsfXU/89j2VikXiTo9j6oV4IFjACNZG6tknNYjOoKtZSBV0A4lJirL0vJLB2k9ZzXp2ySk7U8BFgnJSAjkaPfvT7UckCanQhGOlusiJtZdiqRnhNCu6iaIxJiM8oF1DBTb7vXR6TwaPjdKHQSTNExpO1d8TKQ6VGoe+6QyxHqp5byL+53UTHVx6KRNxoqkgs0VBwqGO4CQc2GeSEs3HhmAimfkrJENsUtAmwqIJAc2fvEhalTJyyui2Wqpf5XEUwCE4AqcAgQtQBzegAZqAgAfwBF7Aq/VoPVtv1vusdcnKZw7AH1gf30aZmvA=AAACD3icbVDLSgMxFM34rPU16tJNsCiuyqRWdFMounFZwT6gM5RMmmlDM5khyQhlmD9w46+4caGIW7fu/BvTdkBtPRA495x7ubnHjzlT2nG+rKXlldW19cJGcXNre2fX3ttvqSiRhDZJxCPZ8bGinAna1Exz2oklxaHPadsfXU/89j2VikXiTo9j6oV4IFjACNZG6tknNYjOoKtZSBV0A4lJirL0vJLB2k9ZzXp2ySk7U8BFgnJSAjkaPfvT7UckCanQhGOlusiJtZdiqRnhNCu6iaIxJiM8oF1DBTb7vXR6TwaPjdKHQSTNExpO1d8TKQ6VGoe+6QyxHqp5byL+53UTHVx6KRNxoqkgs0VBwqGO4CQc2GeSEs3HhmAimfkrJENsUtAmwqIJAc2fvEhalTJyyui2Wqpf5XEUwCE4AqcAgQtQBzegAZqAgAfwBF7Aq/VoPVtv1vusdcnKZw7AH1gf30aZmvA=AAACD3icbVDLSgMxFM34rPU16tJNsCiuyqRWdFMounFZwT6gM5RMmmlDM5khyQhlmD9w46+4caGIW7fu/BvTdkBtPRA495x7ubnHjzlT2nG+rKXlldW19cJGcXNre2fX3ttvqSiRhDZJxCPZ8bGinAna1Exz2oklxaHPadsfXU/89j2VikXiTo9j6oV4IFjACNZG6tknNYjOoKtZSBV0A4lJirL0vJLB2k9ZzXp2ySk7U8BFgnJSAjkaPfvT7UckCanQhGOlusiJtZdiqRnhNCu6iaIxJiM8oF1DBTb7vXR6TwaPjdKHQSTNExpO1d8TKQ6VGoe+6QyxHqp5byL+53UTHVx6KRNxoqkgs0VBwqGO4CQc2GeSEs3HhmAimfkrJENsUtAmwqIJAc2fvEhalTJyyui2Wqpf5XEUwCE4AqcAgQtQBzegAZqAgAfwBF7Aq/VoPVtv1vusdcnKZw7AH1gf30aZmvA=AAACD3icbVDLSgMxFM34rPU16tJNsCiuyqRWdFMounFZwT6gM5RMmmlDM5khyQhlmD9w46+4caGIW7fu/BvTdkBtPRA495x7ubnHjzlT2nG+rKXlldW19cJGcXNre2fX3ttvqSiRhDZJxCPZ8bGinAna1Exz2oklxaHPadsfXU/89j2VikXiTo9j6oV4IFjACNZG6tknNYjOoKtZSBV0A4lJirL0vJLB2k9ZzXp2ySk7U8BFgnJSAjkaPfvT7UckCanQhGOlusiJtZdiqRnhNCu6iaIxJiM8oF1DBTb7vXR6TwaPjdKHQSTNExpO1d8TKQ6VGoe+6QyxHqp5byL+53UTHVx6KRNxoqkgs0VBwqGO4CQc2GeSEs3HhmAimfkrJENsUtAmwqIJAc2fvEhalTJyyui2Wqpf5XEUwCE4AqcAgQtQBzegAZqAgAfwBF7Aq/VoPVtv1vusdcnKZw7AH1gf30aZmvA=

Definitions: Disjoint EventsTwo compound events without a point in common are termed disjoint.

A B

P (A [ B) = P (A) + P (B)AAACAnicbVDLSgMxFL3js9bXqCtxEyxCi1BmRNCNUOvGZQX7gHYomTRtQzOZIckIZWjd+CtuXCji1q9w59+YtrPQ1gMhJ+fcy809fsSZ0o7zbS0tr6yurWc2sptb2zu79t5+TYWxJLRKQh7Kho8V5UzQqmaa00YkKQ58Tuv+4Gbi1x+oVCwU93oYUS/APcG6jGBtpLZ9WMlfj1skjsblArpC5lVAp+YqF9p2zik6U6BF4qYkBykqbfur1QlJHFChCcdKNV0n0l6CpWaE01G2FSsaYTLAPdo0VOCAKi+ZrjBCJ0bpoG4ozREaTdXfHQkOlBoGvqkMsO6reW8i/uc1Y9299BImolhTQWaDujFHOkSTPFCHSUo0HxqCiWTmr4j0scREm9SyJgR3fuVFUjsruk7RvTvPlcppHBk4gmPIgwsXUIJbqEAVCDzCM7zCm/VkvVjv1sesdMlKew7gD6zPHz05lBY=AAACAnicbVDLSgMxFL3js9bXqCtxEyxCi1BmRNCNUOvGZQX7gHYomTRtQzOZIckIZWjd+CtuXCji1q9w59+YtrPQ1gMhJ+fcy809fsSZ0o7zbS0tr6yurWc2sptb2zu79t5+TYWxJLRKQh7Kho8V5UzQqmaa00YkKQ58Tuv+4Gbi1x+oVCwU93oYUS/APcG6jGBtpLZ9WMlfj1skjsblArpC5lVAp+YqF9p2zik6U6BF4qYkBykqbfur1QlJHFChCcdKNV0n0l6CpWaE01G2FSsaYTLAPdo0VOCAKi+ZrjBCJ0bpoG4ozREaTdXfHQkOlBoGvqkMsO6reW8i/uc1Y9299BImolhTQWaDujFHOkSTPFCHSUo0HxqCiWTmr4j0scREm9SyJgR3fuVFUjsruk7RvTvPlcppHBk4gmPIgwsXUIJbqEAVCDzCM7zCm/VkvVjv1sesdMlKew7gD6zPHz05lBY=AAACAnicbVDLSgMxFL3js9bXqCtxEyxCi1BmRNCNUOvGZQX7gHYomTRtQzOZIckIZWjd+CtuXCji1q9w59+YtrPQ1gMhJ+fcy809fsSZ0o7zbS0tr6yurWc2sptb2zu79t5+TYWxJLRKQh7Kho8V5UzQqmaa00YkKQ58Tuv+4Gbi1x+oVCwU93oYUS/APcG6jGBtpLZ9WMlfj1skjsblArpC5lVAp+YqF9p2zik6U6BF4qYkBykqbfur1QlJHFChCcdKNV0n0l6CpWaE01G2FSsaYTLAPdo0VOCAKi+ZrjBCJ0bpoG4ozREaTdXfHQkOlBoGvqkMsO6reW8i/uc1Y9299BImolhTQWaDujFHOkSTPFCHSUo0HxqCiWTmr4j0scREm9SyJgR3fuVFUjsruk7RvTvPlcppHBk4gmPIgwsXUIJbqEAVCDzCM7zCm/VkvVjv1sesdMlKew7gD6zPHz05lBY=AAACAnicbVDLSgMxFL3js9bXqCtxEyxCi1BmRNCNUOvGZQX7gHYomTRtQzOZIckIZWjd+CtuXCji1q9w59+YtrPQ1gMhJ+fcy809fsSZ0o7zbS0tr6yurWc2sptb2zu79t5+TYWxJLRKQh7Kho8V5UzQqmaa00YkKQ58Tuv+4Gbi1x+oVCwU93oYUS/APcG6jGBtpLZ9WMlfj1skjsblArpC5lVAp+YqF9p2zik6U6BF4qYkBykqbfur1QlJHFChCcdKNV0n0l6CpWaE01G2FSsaYTLAPdo0VOCAKi+ZrjBCJ0bpoG4ozREaTdXfHQkOlBoGvqkMsO6reW8i/uc1Y9299BImolhTQWaDujFHOkSTPFCHSUo0HxqCiWTmr4j0scREm9SyJgR3fuVFUjsruk7RvTvPlcppHBk4gmPIgwsXUIJbqEAVCDzCM7zCm/VkvVjv1sesdMlKew7gD6zPHz05lBY=

Probability of a picked card being or

Definitions: Non-disjoint Events

Two compound events without a point in common are termed disjoint.

A B

P (A [ B) = P (A) + P (B)� P (A \ B)AAACEHicbVDLSgMxFL1TX7W+Rl26CRaxRSwzIuhGqHXjsoJ9QDuUTJppQzMPkoxQhvYP3Pgrblwo4talO//GtJ2Fth4IOfece0nucSPOpLKsbyOztLyyupZdz21sbm3vmLt7dRnGgtAaCXkomi6WlLOA1hRTnDYjQbHvctpwBzcTv/FAhWRhcK+GEXV83AuYxwhWWuqYx9XC9bhN4mhcKaIrpKsiOtGXrk7RzMMTr2PmrZI1BVokdkrykKLaMb/a3ZDEPg0U4VjKlm1FykmwUIxwOsq1Y0kjTAa4R1uaBtin0kmmC43QkVa6yAuFPoFCU/X3RIJ9KYe+qzt9rPpy3puI/3mtWHmXTsKCKFY0ILOHvJgjFaJJOqjLBCWKDzXBRDD9V0T6WGCidIY5HYI9v/IiqZ+VbKtk353ny5U0jiwcwCEUwIYLKMMtVKEGBB7hGV7hzXgyXox342PWmjHSmX34A+PzB/SamL8=AAACEHicbVDLSgMxFL1TX7W+Rl26CRaxRSwzIuhGqHXjsoJ9QDuUTJppQzMPkoxQhvYP3Pgrblwo4talO//GtJ2Fth4IOfece0nucSPOpLKsbyOztLyyupZdz21sbm3vmLt7dRnGgtAaCXkomi6WlLOA1hRTnDYjQbHvctpwBzcTv/FAhWRhcK+GEXV83AuYxwhWWuqYx9XC9bhN4mhcKaIrpKsiOtGXrk7RzMMTr2PmrZI1BVokdkrykKLaMb/a3ZDEPg0U4VjKlm1FykmwUIxwOsq1Y0kjTAa4R1uaBtin0kmmC43QkVa6yAuFPoFCU/X3RIJ9KYe+qzt9rPpy3puI/3mtWHmXTsKCKFY0ILOHvJgjFaJJOqjLBCWKDzXBRDD9V0T6WGCidIY5HYI9v/IiqZ+VbKtk353ny5U0jiwcwCEUwIYLKMMtVKEGBB7hGV7hzXgyXox342PWmjHSmX34A+PzB/SamL8=AAACEHicbVDLSgMxFL1TX7W+Rl26CRaxRSwzIuhGqHXjsoJ9QDuUTJppQzMPkoxQhvYP3Pgrblwo4talO//GtJ2Fth4IOfece0nucSPOpLKsbyOztLyyupZdz21sbm3vmLt7dRnGgtAaCXkomi6WlLOA1hRTnDYjQbHvctpwBzcTv/FAhWRhcK+GEXV83AuYxwhWWuqYx9XC9bhN4mhcKaIrpKsiOtGXrk7RzMMTr2PmrZI1BVokdkrykKLaMb/a3ZDEPg0U4VjKlm1FykmwUIxwOsq1Y0kjTAa4R1uaBtin0kmmC43QkVa6yAuFPoFCU/X3RIJ9KYe+qzt9rPpy3puI/3mtWHmXTsKCKFY0ILOHvJgjFaJJOqjLBCWKDzXBRDD9V0T6WGCidIY5HYI9v/IiqZ+VbKtk353ny5U0jiwcwCEUwIYLKMMtVKEGBB7hGV7hzXgyXox342PWmjHSmX34A+PzB/SamL8=AAACEHicbVDLSgMxFL1TX7W+Rl26CRaxRSwzIuhGqHXjsoJ9QDuUTJppQzMPkoxQhvYP3Pgrblwo4talO//GtJ2Fth4IOfece0nucSPOpLKsbyOztLyyupZdz21sbm3vmLt7dRnGgtAaCXkomi6WlLOA1hRTnDYjQbHvctpwBzcTv/FAhWRhcK+GEXV83AuYxwhWWuqYx9XC9bhN4mhcKaIrpKsiOtGXrk7RzMMTr2PmrZI1BVokdkrykKLaMb/a3ZDEPg0U4VjKlm1FykmwUIxwOsq1Y0kjTAa4R1uaBtin0kmmC43QkVa6yAuFPoFCU/X3RIJ9KYe+qzt9rPpy3puI/3mtWHmXTsKCKFY0ILOHvJgjFaJJOqjLBCWKDzXBRDD9V0T6WGCidIY5HYI9v/IiqZ+VbKtk353ny5U0jiwcwCEUwIYLKMMtVKEGBB7hGV7hzXgyXox342PWmjHSmX34A+PzB/SamL8=

Probability of a picked card having the number 3 or

Definitions: Conditional Probability

We will often be interested in events which occur under certain conditions.

Conditional probability measures the effect (if any) of the occurrence of an even A when it is known that even B has occurred in the expt.

E.g. What’s the probability of a card drawn having given that it is a one-eyed jack?

P (X|Y ) := P (XY )P (Y )

AAACBXicbVDLSgMxFL1TX7W+Rl3qIliEdlNmRFAEoejGZQX7oi0lk2ba0ExmSDJCGbtx46+4caGIW//BnX9j2s5CWw8kOTnnXpJ7vIgzpR3n28osLa+srmXXcxubW9s79u5eTYWxJLRKQh7KhocV5UzQqmaa00YkKQ48Tuve8Hri1++pVCwUd3oU0U6A+4L5jGBtpK59WCk0HppFdHGJ2r7EJDH3ZnFsDrN37bxTcqZAi8RNSR5SVLr2V7sXkjigQhOOlWq5TqQ7CZaaEU7HuXasaITJEPdpy1CBA6o6yXSKMTo2Sg/5oTRLaDRVf3ckOFBqFHimMsB6oOa9ifif14q1f95JmIhiTQWZPeTHHOkQTSJBPSYp0XxkCCaSmb8iMsAmDG2Cy5kQ3PmRF0ntpOQ6Jff2NF++SuPIwgEcQQFcOIMy3EAFqkDgEZ7hFd6sJ+vFerc+ZqUZK+3Zhz+wPn8AOzOWfA==AAACBXicbVDLSgMxFL1TX7W+Rl3qIliEdlNmRFAEoejGZQX7oi0lk2ba0ExmSDJCGbtx46+4caGIW//BnX9j2s5CWw8kOTnnXpJ7vIgzpR3n28osLa+srmXXcxubW9s79u5eTYWxJLRKQh7KhocV5UzQqmaa00YkKQ48Tuve8Hri1++pVCwUd3oU0U6A+4L5jGBtpK59WCk0HppFdHGJ2r7EJDH3ZnFsDrN37bxTcqZAi8RNSR5SVLr2V7sXkjigQhOOlWq5TqQ7CZaaEU7HuXasaITJEPdpy1CBA6o6yXSKMTo2Sg/5oTRLaDRVf3ckOFBqFHimMsB6oOa9ifif14q1f95JmIhiTQWZPeTHHOkQTSJBPSYp0XxkCCaSmb8iMsAmDG2Cy5kQ3PmRF0ntpOQ6Jff2NF++SuPIwgEcQQFcOIMy3EAFqkDgEZ7hFd6sJ+vFerc+ZqUZK+3Zhz+wPn8AOzOWfA==AAACBXicbVDLSgMxFL1TX7W+Rl3qIliEdlNmRFAEoejGZQX7oi0lk2ba0ExmSDJCGbtx46+4caGIW//BnX9j2s5CWw8kOTnnXpJ7vIgzpR3n28osLa+srmXXcxubW9s79u5eTYWxJLRKQh7KhocV5UzQqmaa00YkKQ48Tuve8Hri1++pVCwUd3oU0U6A+4L5jGBtpK59WCk0HppFdHGJ2r7EJDH3ZnFsDrN37bxTcqZAi8RNSR5SVLr2V7sXkjigQhOOlWq5TqQ7CZaaEU7HuXasaITJEPdpy1CBA6o6yXSKMTo2Sg/5oTRLaDRVf3ckOFBqFHimMsB6oOa9ifif14q1f95JmIhiTQWZPeTHHOkQTSJBPSYp0XxkCCaSmb8iMsAmDG2Cy5kQ3PmRF0ntpOQ6Jff2NF++SuPIwgEcQQFcOIMy3EAFqkDgEZ7hFd6sJ+vFerc+ZqUZK+3Zhz+wPn8AOzOWfA==AAACBXicbVDLSgMxFL1TX7W+Rl3qIliEdlNmRFAEoejGZQX7oi0lk2ba0ExmSDJCGbtx46+4caGIW//BnX9j2s5CWw8kOTnnXpJ7vIgzpR3n28osLa+srmXXcxubW9s79u5eTYWxJLRKQh7KhocV5UzQqmaa00YkKQ48Tuve8Hri1++pVCwUd3oU0U6A+4L5jGBtpK59WCk0HppFdHGJ2r7EJDH3ZnFsDrN37bxTcqZAi8RNSR5SVLr2V7sXkjigQhOOlWq5TqQ7CZaaEU7HuXasaITJEPdpy1CBA6o6yXSKMTo2Sg/5oTRLaDRVf3ckOFBqFHimMsB6oOa9ifif14q1f95JmIhiTQWZPeTHHOkQTSJBPSYp0XxkCCaSmb8iMsAmDG2Cy5kQ3PmRF0ntpOQ6Jff2NF++SuPIwgEcQQFcOIMy3EAFqkDgEZ7hFd6sJ+vFerc+ZqUZK+3Zhz+wPn8AOzOWfA==

When the two events are independent we have P (X|Y )P (Y ) = P (Y |X)P (Y )AAACAnicbVDLSgMxFL1TX7W+Rl2Jm2AR6qbMiKAboejG5Qi2HWmHkknTNjTzIMkIZVrc+CtuXCji1q9w59+YaWehrQeSnJxzL8k9fsyZVJb1bRSWlldW14rrpY3Nre0dc3evIaNEEFonEY+E62NJOQtpXTHFqRsLigOf06Y/vM785gMVkkXhnRrF1AtwP2Q9RrDSUsc8cCru+P4EORW9XWbH2J3dOmbZqlpToEVi56QMOZyO+dXuRiQJaKgIx1K2bCtWXoqFYoTTSamdSBpjMsR92tI0xAGVXjodYYKOtdJFvUjoFSo0VX93pDiQchT4ujLAaiDnvUz8z2slqnfhpSyME0VDMnuol3CkIpTlgbpMUKL4SBNMBNN/RWSABSZKp1bSIdjzIy+SxmnVtqr27Vm5dpXHUYRDOIIK2HAONbgBB+pA4BGe4RXejCfjxXg3PmalBSPv2Yc/MD5/APwhk/I=AAACAnicbVDLSgMxFL1TX7W+Rl2Jm2AR6qbMiKAboejG5Qi2HWmHkknTNjTzIMkIZVrc+CtuXCji1q9w59+YaWehrQeSnJxzL8k9fsyZVJb1bRSWlldW14rrpY3Nre0dc3evIaNEEFonEY+E62NJOQtpXTHFqRsLigOf06Y/vM785gMVkkXhnRrF1AtwP2Q9RrDSUsc8cCru+P4EORW9XWbH2J3dOmbZqlpToEVi56QMOZyO+dXuRiQJaKgIx1K2bCtWXoqFYoTTSamdSBpjMsR92tI0xAGVXjodYYKOtdJFvUjoFSo0VX93pDiQchT4ujLAaiDnvUz8z2slqnfhpSyME0VDMnuol3CkIpTlgbpMUKL4SBNMBNN/RWSABSZKp1bSIdjzIy+SxmnVtqr27Vm5dpXHUYRDOIIK2HAONbgBB+pA4BGe4RXejCfjxXg3PmalBSPv2Yc/MD5/APwhk/I=AAACAnicbVDLSgMxFL1TX7W+Rl2Jm2AR6qbMiKAboejG5Qi2HWmHkknTNjTzIMkIZVrc+CtuXCji1q9w59+YaWehrQeSnJxzL8k9fsyZVJb1bRSWlldW14rrpY3Nre0dc3evIaNEEFonEY+E62NJOQtpXTHFqRsLigOf06Y/vM785gMVkkXhnRrF1AtwP2Q9RrDSUsc8cCru+P4EORW9XWbH2J3dOmbZqlpToEVi56QMOZyO+dXuRiQJaKgIx1K2bCtWXoqFYoTTSamdSBpjMsR92tI0xAGVXjodYYKOtdJFvUjoFSo0VX93pDiQchT4ujLAaiDnvUz8z2slqnfhpSyME0VDMnuol3CkIpTlgbpMUKL4SBNMBNN/RWSABSZKp1bSIdjzIy+SxmnVtqr27Vm5dpXHUYRDOIIK2HAONbgBB+pA4BGe4RXejCfjxXg3PmalBSPv2Yc/MD5/APwhk/I=AAACAnicbVDLSgMxFL1TX7W+Rl2Jm2AR6qbMiKAboejG5Qi2HWmHkknTNjTzIMkIZVrc+CtuXCji1q9w59+YaWehrQeSnJxzL8k9fsyZVJb1bRSWlldW14rrpY3Nre0dc3evIaNEEFonEY+E62NJOQtpXTHFqRsLigOf06Y/vM785gMVkkXhnRrF1AtwP2Q9RrDSUsc8cCru+P4EORW9XWbH2J3dOmbZqlpToEVi56QMOZyO+dXuRiQJaKgIx1K2bCtWXoqFYoTTSamdSBpjMsR92tI0xAGVXjodYYKOtdJFvUjoFSo0VX93pDiQchT4ujLAaiDnvUz8z2slqnfhpSyME0VDMnuol3CkIpTlgbpMUKL4SBNMBNN/RWSABSZKp1bSIdjzIy+SxmnVtqr27Vm5dpXHUYRDOIIK2HAONbgBB+pA4BGe4RXejCfjxXg3PmalBSPv2Yc/MD5/APwhk/I=

P (XY ) = P (X)P (Y )AAAB+XicbVDLSsNAFL2pr1pfUZduBovQbkoigm6EohuXEeyLNpTJdNIOnUzCzKRQQv/EjQtF3Pon7vwbp20W2nrgcg/n3MvcOUHCmdKO820VNja3tneKu6W9/YPDI/v4pKniVBLaIDGPZTvAinImaEMzzWk7kRRHAaetYHw/91sTKhWLxZOeJtSP8FCwkBGsjdS3ba/S7lTRLTK96lU61b5ddmrOAmiduDkpQw6vb3/1BjFJIyo04Viprusk2s+w1IxwOiv1UkUTTMZ4SLuGChxR5WeLy2fowigDFMbSlNBoof7eyHCk1DQKzGSE9UitenPxP6+b6vDGz5hIUk0FWT4UphzpGM1jQAMmKdF8aggmkplbERlhiYk2YZVMCO7ql9dJ87LmOjX38apcv8vjKMIZnEMFXLiGOjyABw0gMIFneIU3K7NerHfrYzlasPKdU/gD6/MHoZyRDQ==AAAB+XicbVDLSsNAFL2pr1pfUZduBovQbkoigm6EohuXEeyLNpTJdNIOnUzCzKRQQv/EjQtF3Pon7vwbp20W2nrgcg/n3MvcOUHCmdKO820VNja3tneKu6W9/YPDI/v4pKniVBLaIDGPZTvAinImaEMzzWk7kRRHAaetYHw/91sTKhWLxZOeJtSP8FCwkBGsjdS3ba/S7lTRLTK96lU61b5ddmrOAmiduDkpQw6vb3/1BjFJIyo04Viprusk2s+w1IxwOiv1UkUTTMZ4SLuGChxR5WeLy2fowigDFMbSlNBoof7eyHCk1DQKzGSE9UitenPxP6+b6vDGz5hIUk0FWT4UphzpGM1jQAMmKdF8aggmkplbERlhiYk2YZVMCO7ql9dJ87LmOjX38apcv8vjKMIZnEMFXLiGOjyABw0gMIFneIU3K7NerHfrYzlasPKdU/gD6/MHoZyRDQ==AAAB+XicbVDLSsNAFL2pr1pfUZduBovQbkoigm6EohuXEeyLNpTJdNIOnUzCzKRQQv/EjQtF3Pon7vwbp20W2nrgcg/n3MvcOUHCmdKO820VNja3tneKu6W9/YPDI/v4pKniVBLaIDGPZTvAinImaEMzzWk7kRRHAaetYHw/91sTKhWLxZOeJtSP8FCwkBGsjdS3ba/S7lTRLTK96lU61b5ddmrOAmiduDkpQw6vb3/1BjFJIyo04Viprusk2s+w1IxwOiv1UkUTTMZ4SLuGChxR5WeLy2fowigDFMbSlNBoof7eyHCk1DQKzGSE9UitenPxP6+b6vDGz5hIUk0FWT4UphzpGM1jQAMmKdF8aggmkplbERlhiYk2YZVMCO7ql9dJ87LmOjX38apcv8vjKMIZnEMFXLiGOjyABw0gMIFneIU3K7NerHfrYzlasPKdU/gD6/MHoZyRDQ==AAAB+XicbVDLSsNAFL2pr1pfUZduBovQbkoigm6EohuXEeyLNpTJdNIOnUzCzKRQQv/EjQtF3Pon7vwbp20W2nrgcg/n3MvcOUHCmdKO820VNja3tneKu6W9/YPDI/v4pKniVBLaIDGPZTvAinImaEMzzWk7kRRHAaetYHw/91sTKhWLxZOeJtSP8FCwkBGsjdS3ba/S7lTRLTK96lU61b5ddmrOAmiduDkpQw6vb3/1BjFJIyo04Viprusk2s+w1IxwOiv1UkUTTMZ4SLuGChxR5WeLy2fowigDFMbSlNBoof7eyHCk1DQKzGSE9UitenPxP6+b6vDGz5hIUk0FWT4UphzpGM1jQAMmKdF8aggmkplbERlhiYk2YZVMCO7ql9dJ87LmOjX38apcv8vjKMIZnEMFXLiGOjyABw0gMIFneIU3K7NerHfrYzlasPKdU/gD6/MHoZyRDQ==

Or