A drawer holds 4 red hats and 4 blue hats. What is the...

Expert replies

AAPL: Moderator; Posts: 2636; Joined: Sun Oct 29, 2017 2:08 pmFollowed by: 2 members

by AAPL » Wed Nov 29, 2017 6:51 pm

A drawer holds 4 red hats and 4 blue hats. What is the probability of getting exactly three red hats or exactly three blue hats when taking out 4 hats randomly out of the drawer and returning each hat before taking out the next one?

A. 1/8
B. 1/4
C. 1/2
D. 3/8
E. 7/12

The OA is C.

Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.

Like this post

Source: Beat The GMAT — Problem Solving | Wed Nov 29, 2017 6:51 pm

GMAT/MBA Expert

[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Wed Nov 29, 2017 7:03 pm

Hi swerve,

We're told that a drawer holds 4 red hats and 4 blue hats. We're asked for the probability of getting exactly three red hats or exactly three blue hats when taking out 4 hats randomly out of the drawer and returning each hat before taking out the next one. We can answer this question with a bit of note-taking and some basic arithmetic. To start, since we're putting each hat back into the drawer after pulling one out, the fact that there are 4 red hats and 4 blue hats is irrelevant. As long as there are an equal number of both colors, the math will be exactly the same (meaning that there could be 1 red hat and 1 blue hat).

Since there is an equal chance of pulling either of the two colors on each 'pull', there are (2)(2)(2)(2) = 16 possible outcomes

Given the 'goal', there are only a limited number of possible outcomes that 'fit' what we're looking for:
RRRB
RRBR
RBRR
BRRR

BBBR
BBRB
BRBB
RBBB

8 of the 16 = 1/2 of the outcomes give us what we're after.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

Like this post

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Nov 30, 2017 4:00 am

AAPL wrote:A drawer holds 4 red hats and 4 blue hats. What is the probability of getting exactly three red hats or exactly three blue hats when taking out 4 hats randomly out of the drawer and returning each hat before taking out the next one?

A. 1/8
B. 1/4
C. 1/2
D. 3/8
E. 7/12
[/quote

P(exactly n times) = P(one way) * total possible ways.

Since 4 of the 8 hats are red, and 4 of the 8 hats are blue:
P(R) = 4/8 = 1/2.
P(B) = 4/8 = 1/2.

P(one way):
One way to select exactly 3 blue hats:
RBBB.
P(R on the 1st pick) = 1/2.
P(B on the 2nd pick) = 1/2.
P(B on the 3rd pick) = 1/2.
P(B on the 4th pick) = 1/2.
Since we want all of these events to happen, we MULTIPLY the fractions:
1/2 * 1/2 * 1/2 * 1/2 = 1/16.

Total possible ways:
RBBB is only ONE WAY to get exactly 3 blue hats.
Now we must account for ALL OF THE WAYS to get exactly 3 blue hats.
Any arrangement of the letters RBBB represents one way to get exactly 3 blue hats.
Thus, to account for ALL OF THE WAYS to get exactly 3 blue hats, the result above must be multiplied by the number of ways to arrange the letters RBBB.
Number of ways to arrange 4 elements = 4!.
But when an arrangement includes IDENTICAL elements, we must divide by the number of ways each set of identical elements can be ARRANGED.
The reason:
When the identical elements swap positions, the arrangement doesn't change.
Here, we must divide by 3! to account for the three identical B's:
4!/(3!) = 4.

Multiplying the results above, we get:
P(exactly 3 blue hats) = 1/16 * 4 = 1/4.

By extension:
P(exactly 3 red hats) = 1/4.

Since a favorable outcome will be yielded by selecting exactly 3 blue hats OR exactly 3 red hats, we ADD the fractions in blue:
P(3 hats of the same color) = P(exactly 3 blue hats) + P(exactly 3 red hats) = 1/4 + 1/4 = 1/2.

The correct answer is C.

More practice:
https://www.beatthegmat.com/select-exact ... 88786.html
https://www.beatthegmat.com/probability- ... 14250.html
https://www.beatthegmat.com/a-single-par ... 28342.html
https://www.beatthegmat.com/at-a-blind-t ... 20058.html
https://www.beatthegmat.com/rain-check-t79099.html
https://www.beatthegmat.com/probability-t227448.html

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Like this post

GMAT/MBA Expert

Scott@TargetTestPrep: GMAT Instructor; Posts: 8139; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Sat Oct 05, 2019 3:54 pm

AAPL wrote:A drawer holds 4 red hats and 4 blue hats. What is the probability of getting exactly three red hats or exactly three blue hats when taking out 4 hats randomly out of the drawer and returning each hat before taking out the next one?

A. 1/8
B. 1/4
C. 1/2
D. 3/8
E. 7/12

The OA is C.

Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.

First, let's determine the probability of selecting exactly 3 red hats (and 1 blue hat).

P(R-R-R-B) = (1/2)^4 = 1/16

Since R-R-R-B can be arranged in 4!/3! = 4 ways, the overall probability is 4/16 = 1/4.

Since we have the same number of red and blue hats the probability of selecting exactly 3 blue hats (and 1 red hat) is also 1/4.

Thus, the probability of selecting exactly 3 red hats or 3 blue hats is 1/4 + 1/4 = 2/4 = 1/2.

Answer: C

Scott Woodbury-Stewart
Founder and CEO
[email protected]

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

560 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review