As a test for extrasensory perception (ESP), eight cards, four black and four white, are shuffled, and then a person looks at each card. In another room, the ESP subject attempts to guess whether the card is black or white. The ESP subject must guess black four times and white four times. If the ESP subject has no extrasensory perception, what is the probability that the subject will correctly name six of the eight cards? Round to the nearest thousandth.
A)0.014
B)0.229
C)0.450
D)0.667
E)0.039

Question

Quizplus · Accepted Answer

There are a total of ${8\choose6}=28$ ways in which the ESP subject can correctly guess six of the eight cards. 
For each of these 28 ways, there are $2^2=4$ ways in which the remaining two cards can be guessed incorrectly. 
Therefore, the total number of ways in which the ESP subject can guess 6 out of 8 correctly is $28	imes4=112$. 
Out of the total $2^8=256$ possible outcomes, only 112 will result in the ESP subject guessing 6 of the 8 cards correctly without ESP. 
Thus, the probability is $\dfrac{112}{256} \approx 0.229$. 
Therefore, the best choice is $	extbf{(B)}.$

As a Test for Extrasensory Perception (ESP), Eight Cards, Four