GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 21 Mar 2019, 13:02

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Three points are randomly chosen on the circumference of a given circl

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

 
GMATH Teacher
User avatar
G
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 905
Three points are randomly chosen on the circumference of a given circl  [#permalink]

Show Tags

New post 18 Feb 2019, 13:58
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

29% (02:13) correct 71% (01:54) wrong based on 34 sessions

HideShow timer Statistics

GMATH practice exercise (Quant Class 19)

Three points are randomly chosen on the circumference of a given circle. What is the probability that the center of the circle lies inside the triangle whose vertices are at the three points?

(A) 1/3
(B) 1/4
(C) 1/5
(D) 2/5
(E) 2/7

Hint: the problem was created based on another given in the link hidden below
https://gmatclub.com/forum/three-points-are-chosen-independently-an-at-random-on-the-circumferenc-205189.html

_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

GMATH Teacher
User avatar
G
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 905
Re: Three points are randomly chosen on the circumference of a given circl  [#permalink]

Show Tags

New post 19 Feb 2019, 12:29
fskilnik wrote:
GMATH practice exercise (Quant Class 19)

Three points are randomly chosen on the circumference of a given circle. What is the probability that the center of the circle lies inside the triangle whose vertices are at the three points?

(A) 1/3
(B) 1/4
(C) 1/5
(D) 2/5
(E) 2/7

We will refer to the link given in the hint: https://gmatclub.com/forum/three-points ... 05189.html

Image

FOCUS: P(center inside the triangle)

We may consider (without loss of generality) that the first point (blue) was chosen at the position shown in the first figure and that the second point (red) was chosen (without loss of generality) in the upper semicircle (red arc) also presented in the first figure.

In the second figure, we show one possibility of the first two points, to illustrate the fact that we will have a favorable scenario if and only if the third point (green) is chosen in the arc in green, obtained by two lines, each one defined by the center and one of the two first points.

Considering exactly the same reasoning presented in the exercise mentioned in the hint, let´s consider the two extremal cases: when the first two points ("almost") coincide and when the first two points are ("almost") diametrically opposites. The corresponding favorable probabilities are 0 and 1/2, respectively, hence (following the same rationale presented there) our FOCUS is their average: (0+1/2)/2 = 1/4.

The correct answer is (B).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

POST-MORTEM: it is possible to prove that the probabilities of having the first and second points coincidental or diametrically opposites are both ZERO (not approximately zero), therefore there is no trouble omitting the words "almost" presented in parentheses. A subtle consequence: the answer obtained is not approximately right, it is exactly right. (Details are out-of-GMAT´s-scope.)
_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

GMAT Club Bot
Re: Three points are randomly chosen on the circumference of a given circl   [#permalink] 19 Feb 2019, 12:29
Display posts from previous: Sort by

Three points are randomly chosen on the circumference of a given circl

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.