psm12se wrote:At a blind taste competition a contestant is offered 3 cups of each of the 3 samples of tea in a random arrangement of 9 marked cups. If each contestant tastes 4 different cups of tea, what is the probability that a contestant does not taste all of the samples?
A. 1/12
B. 5/14
C. 4/9
D. 1/2
E. 2/3
This question can be solved using the complement.
That is, P(Event A happening) = 1 - P(Event A
not happening)
So, here we get: P(contestant does not taste all 3 samples) = 1 -
P(contestant DOES taste all 3 samples)
P(contestant DOES taste all 3 samples)
For this event to occur, the contestant must taste 2 cups of one sample, 1 cup from another sample, and 1 cup from another sample.
Let's take the task of tasting all 3 samples and break it into STAGES.
Stage 1: Select the tea that will be tasted twice. There are 3 types of tea, so stage 1 can be completed in
3 ways.
Stage 2: Choose 2 cups to taste from tea selected in stage 1. Since the order in which we select the 2 cups does not matter, we can use combinations. We can select 2 cups from 3 cups in 3C2 ways(
= 3 ways).
Stage 3: From one of the two remaining (untasted) teas, select 1 cup to taste. There are 3 cups, so stage 3 can be completed in
3 ways.
Stage 4: Select 1 cup from the last remaining (untasted) tea. There are 3 cups, so stage 4 can be completed in
3 ways.
By the Fundamental Counting Principle (FCP), we can complete all 4 stages in
(3)(3)(3)(3) ways (=
81 ways)
The TOTAL number of ways to select 4 cups from 9 cups = 9C4 =
126
So,
P(contestant DOES taste all 3 samples) = 81/126 = 9/14
This means that P(contestant does not taste all 3 samples) = 1 -
9/14
= [spoiler]5/14[/spoiler]
=
B
Cheers,
Brent
Aside: For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775
Aside: If anyone is interested, we have a free video on calculating combinations (like 3C2 and 9C4) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789
