Question 283185
Pick a card: Mike and Dave play the following game. Mike picks a card from a deck of cards. If he selects a heart, Dave gives him $5. If not, he gives Dave $2.
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Mike's net winnings are 

 +$5 if he selects a heart or -$2 if he doesn't select a heart

Mike's expectation = 

(net winnings if he gets a heart)*(probability of getting a heart)
                    
                   plus

(net winnings if he doesn't get a heart)*(prob of not getting a heart)


Probability of Mike getting a heart = {{{13/52}}} or {{{1/4}}}
Probability of Mike NOT getting a heart = {{{39/52}}} or {{{3/4}}}

So,

Mike's expectation = {{{("+$5")(1/4)+("-$2")(3/4)="$5"/4-"$6"/4="-$1"/4="-$.25"}}}

If they continue to play the game many times, in the end 
Mike will have averaged LOSING 25 cents per game.

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Dave's net winnings are 

 -$5 if Mike selects a heart or +$2 if he doesn't select a heart

Dave's expectation = 

(net winnings if Mike gets a heart)*(probability of Mike getting a heart)
                    
                   plus

(net winnings if Mike doesn't get a heart)*(prob of Mike not getting a heart)


Probability of Mike getting a heart = {{{13/52}}} or {{{1/4}}}
Probability of Mike NOT getting a heart = {{{39/52}}} or {{{3/4}}}

So,

Dave's expectation = {{{("-$5")(1/4)+("+$2")(3/4)="-$5"/4+"$6"/4="$1"/4="$.25"}}}

If they continue to play the game many times, in the end 
Dave will have averaged WINNING 25 cents per game.

Edwin</pre>