) Suppose I ask you to pick any four cards at random from a deck of 52, without replacement, and bet that if at least one of the four is a face card (i. e., Jack, Queen, or King), you pay me one dollar; otherwise I pay you one dollar. Should you take the bet? Why? (Hint: See how the probability of this event compares to 50%. If this is too hard, try it with replacement first.) (b) What if the bet involves picking three cards at random instead of four? Should you take the bet then? Why?

Answer & Explanation:

a) Total ways to select 4 cards out of 52

= 52 C 4 = 270725

Total ways to select 4 cards out of 40 other (faceless) cards

= 40 C 4 = 91390

P( faceless cards ) = 91390 / 270725

= 0.3376

P( minimum 1 face ) = 1 - P( faceless cards )

= 1 - 0.3376

= 0.6624

You should take the bet because probability of winning is greater than 50%.

b) Total ways to select 3 cards out of 52

= 52 C 3 = 22100

Total ways to select 4 cards out of 40 other (faceless) cards

= 40 C 3 = 9880

P( faceless cards ) = 9880 / 22100

= 0.4471

P( minimum 1 face ) = 1 - P( faceless cards )

= 1 - 0.4471

= 0.5529

You should take the bet because probability of winning is greater than 50%.