doomedcat wrote:
ida had 5 cards with matching envelopes - same in design, different in color. She removed the cards from all envelopes and randomly put them back. What is the probability that exactly one card got into the matching envelope?A \(\frac{3}{16}\)B \(\frac{5}{8}\)C \(\frac{3}{8}\)D \(\frac{1}{4}\)E \(\frac{1}{2}\)Source :
Experts Global Answer: D
The question asks for "exactly one card into the right envelope", hence the combination of right & wrong cards will look like
R W W W W
Assuming the first card is put in right envelope, since there is only one right card out of the 5 for the first envelope, the probability is 1/5
For the second envelope, there are 3 wrong cards out of the 4 cards left, hence the probability for the Second card in wrong envelope is 3/4
For the third envelope, there are 2 wrong cards out of the 3 cards left, hence the probability for the third card in wrong envelope is 2/3
For the fourth envelope, there is 1 wrong card out of the 2 cards left, hence the probability for the fourth card in wrong envelope is 1/2
For the fifth envelope, since only one card is left now & a wrong one at that, hence the probability for the fifth card in wrong envelope is 1
Moreover the arrangement R W W W W can be arranged in 5!/4! ways
Hence the probability of exactly one right card \(= (1/5)*(3/4)*(2/3)*(1/2)*(1) * (5!/4!) = 1/4\)