Hi Bunuel, beautiful question.
Question - In approaching this, I had 6 slots, 2 for each couple of husband and wife
First slot, husband has 1/9 probability of choosing a specific dessert, wife also has 1/9 of choosing that same dessert.
Why is the dessert the husband chooses 1 instead of 1/9?
Thank you for your time. I’ve been doing GMATClub quant tests and if anyone is reading this, don’t hesitate, buy it and study it. It is an amazing resource and the solutions and then discussions are amazing. Thank you for all you do Bunuel and team GMATClub!
Bunuel wrote:
Official Solution:
At a restaurant, three couples are preparing to order dessert. If there are 9 different desserts on the menu, what is the probability that the husband and wife of each couple will order the same dessert? Note that while each couple is required to order the same dessert, this dessert does not need to match the one ordered by the other couples.
A. \(\frac{6}{9^3}\)
B. \(\frac{3}{9^3}\)
C. \(\frac{1}{9^3}\)
D. \(\frac{1}{9^6}\)
E. \(\frac{6}{9^6}\)
For each couple, there's a 1/9 chance the wife will order the same dessert as her husband. With three couples, the combined probability is:
\(\frac{1}{9} *\frac{1}{9} *\frac{1}{9} = \frac{1}{9^3} \).
Answer: C
A husband can choose ANY dessert from the 9, so the probability of that is 1. The probability that his wife will choose the same dessert is 1/9.
Or think about this in this way: the husband has a 1/9 probability of choosing some specific desserts, say pavlova, his wife has a 1/9 chance to also choose pavlova. However, we have 9 different desserts and for each, we'd have the same probability of 1/9*1/9, making the overall probability that the husband and wife will choose the same dessert is 9*1/9*1/9 = 1/9.