Information Measures (graded).txt

Suppose we have have two coins: one � fair� coin, where p(head) p(head) = p(tails) p(tails) = .5; and an �unfair� coin where p(heads) p(heads) does does not equal p(tails). p(tails). Which Which coin has a larger larger entropy prior to oserving the outco!e" #he fair coin

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$f you roll one fair dice (%&sided), what is its entropy efore the result is oserved" '.' its

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'.5* its

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$f your friend pic+s one nu!er etween -- to 5---, under the strategy used in video ntropy of a /uessing /a!e, what is the !ai!u! nu!er of questions you need to as+ to find out that nu!er"  '

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0se the �$nfor!ation $nfor!ation /ain 1alculator 1alculator� spreadsheet spreadsheet to calculate calculate the the 21onditio 21onditional nal ntropy2 3(46) given a = -.7, c = -.5, e = -.. $nfor!ation /ain 1alculator.ls -.8- its -.89 its .*9 its

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n the the �$nfor!ation $nfor!ation /ain 1alculator 1alculator� spreadsheet, spreadsheet, given a = -., c = -.', suppose now we also +now that 3(4,6) = 3(4)  3(6). What is the
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/iven a = -.', c = -.5 on the $nfor!ation /ain 1alculator Spreadsheet suppose now we also +now the true positive rate is -.*. What is the utual $nfor!ation" $nfor!ation /ain 1alculator.ls -.-* its

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1onsider the onty 3all prole!, ut instead of the usual  doors, assu!e there are 5 doors to choose fro!. 6ou first choose door >. onty opens doors >' and >. What is the new proaility that there is a pri?e ehind door >7" -.' -.5 -.7

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@gain, consider the onty 3all prole!, ut with 5 doors to choose fro! instead of . 6ou pic+ door >, and onty opens ' of the other 7 doors. 3ow !any its of infor!ation are co!!unicated to you y onty when you oserve which two doors he opens" .5' its

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'.' its

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-.*- its

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A stands for � the coin is fair� , BA stands for � the coin is croo+ed� . #he p(heads  A) = -.5, and p(heads  BA) = -.7. 6our friend tells you that he often tests people to see if they can guess whether he is using the fair coin or the croo+ed coin, ut that he is careful to use the croo+ed coin 9-C of the ti!e. 3e tosses the coin once and it co!es up heads. What is your new est esti!ate of the proaility that the coin he
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-.5

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Suppose you are given either a fair dice or an unfair dice (%&sided). 6ou have no asis for considering either dice !ore li+ely efore you roll it and oserve an outco!e. Dor the fair dice, the chance of oserving � � is E%. Dor the unfair dice, the chance of oserving � � is E. @fter rolling the un+nown dice, you oserve the outco!e to e . What is the new proaility that the die you rolled is fair" -.-*

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-.

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