Suppose that you randomly draw one card from a standard deck of 52 cards. After writing down which card was drawn, you place the card back in the deck, shuffle

Question

Suppose that you randomly draw one card from a standard deck of 52 cards. After writing down which card was drawn, you place the card back in the deck, shuffle the deck, and draw another card. You repeat this process until you have drawn 12 cards in all. What is the probability of drawing at least 8 hearts?

Answer 1

It is binomial probability.

 

There are 13 hearts in the deck of 52.

So the probability of drawing a heart ( a success)

is p=13/52 = 1/4 = 0.25

 

So the probability of success is p=0.25

The probability of failure is 1 - p = 0.75

 

We want 8 or more hearts (successes) in 12 trials

 

The formula for binomial probability is

  SUM  [ (12 choose x) * (0.25)^x * (0.75)^(12-x)]  ,   x = 8,9,10,11,12

 

 

The unlikely probability is calculated via MS Excel in the table below

 

X           12 choose X               0.25^x               0.75^(12-x)         Prob.
8                 495                  1.52588E-05            0.31640625   0.002389848
9                 220                  3.8147E-06              0.421875      0.000354052
10                66                    9.53674E-07               0.5625         3.54052E-05
11                12                     2.38419E-07               0.75           2.14577E-06
12                 1                      5.96046E-08              1                  5.96046E-08

                                                                   TOTAL ------>         0.00278151

 

Not even 0.3% chance