Smith is given a choice between two sealed envelopes. one of them contains twice asmuch money as the other. since the money is in the form of checks payable to her, she cannot tell the envelopes apart before opening them. she picks one and finds a check for xamount of money. she is then given the choice to switch envelopes. now she reasons as so: “if i was lucky, then the other envelope contains ½ x; but if i was unlucky, then it contains 2x. each these scenarios has probability ½. so the expected utility, if i switch, is ½ × ½ x+ ½ × 2x= 5x/4. so i should switch.” yet this would meanthat whichever envelope one chooses in this situation, one should expect the other to contain more money. where does smith’s reasoning go wrong?

Given: ad¯¯¯¯¯ is an altitude. prove: ab2+ac2=cb2 right triangle a b c with right angle a. point d lies on side b c and segment a d is drawn. angle a d c is a right angle. drag and drop a reason into each box to correctly complete the two-column proof. statement reason ad¯¯¯¯¯ is an altitude, and ∠bac is a right angle. given ∠adb and ∠adc are right angles. definition of altitude ∠bac≅∠bda ? ∠bac≅∠adc ? ∠b≅∠b ? ∠c≅∠c reflexive property of congruence △abc∼△dba ? △abc∼△dac aa similarity postulate abbd=cbab ? ab2=(cb)(bd) cross multiply and simplify. acdc=cbac polygon similarity postulate ac2=(cb)(dc) cross multiply and simplify. ab2+ac2=ab2+(cb)(dc) addition property of equality ab2+ac2=(cb)(bd)+(cb)(dc) substitution property of equality ab2+ac2=(cb)(bd+dc) ? bd+dc=cb segment addition postulate ab2+ac2=cb2 substitution property of equality