BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem
TTP OnDemand GMAT Video Course
Featured Course

Target Test Prep GMAT OnDemand

Scott Woodbury-Stewart’s private virtual classroom — 400 hours of master-class video lessons for the GMAT Focus Edition.

  • 715+ score guarantee — highest in the industry (99th percentile)
  • 52 chapters · 1,500+ lessons · 4,000+ practice questions
  • 400 hours of video · 1,500+ instructor-led HD smartboard lessons
  • 300,000+ students accepted to Harvard, Stanford, Wharton, Booth & Sloan & more
  • 24/7 live support + weekly Zoom office hours with GMAT instructors
  • TTP AI Assist — 24/7 AI-powered virtual tutor for instant help
  • 1,200+ flashcards + AI-powered study assistant & daily calendar
  • OnDemand, LiveTeach & GMAT Bootcamp formats available
  • Also: GRE, SAT Math & Executive Assessment courses
  • MBA Admissions Consulting now available
  • 🏆 2025 EdTech Breakthrough Award: Test Prep Solution Provider of the Year
  • 200,000+ students served
  • 5-day free trial — $0 to start, no auto-billing, cancel anytime
★★★★★ 5.0 (559 reviews)
130-pt guarantee
$0 to start then $127/mo
Start Free 5-Day Trial Read 559 Reviews →

Prob of at least 3 heads in 5 coin tosses

Expert replies
Post new topic Post Reply
Previous Topic Next Topic
Source: — Problem Solving |
User avatar
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 Oct 07, 2010 11:32 am
joannabanana wrote:If the coin is tossed 5 times, what is the probability that at least 3 out of 5 times it will show heads?


[spoiler]ans = 1/2[/spoiler]
When we flip a coin 5 times, we get either:

Heads 3 times, tails 2 times
Heads 4 times, tails 1 time
Heads 5 times, tails 0 times

OR

Tails 3 times, heads 2 times
Tails 4 times, heads 1 time
Tails 5 times, heads 0 times

Thus P(heads at least 3 times) = P(tails at least 3 times) = 1/2.
Last edited by GMATGuruNY on Thu Oct 07, 2010 12:41 pm, edited 2 times in total.
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
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
Master | Next Rank: 500 Posts
Posts: 118
Joined: Sun Sep 05, 2010 10:21 am
Location: Canada
Thanked: 5 times
GMAT Score:730

by joannabanana » Thu Oct 07, 2010 11:39 am
Oh wow, that's really clever!

I was looking for a more formula-type answer, but this is way better. I doubt I'd be able to come up with something like this myself though, especially while writing the test. But I guess that's what these forums are for :)
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
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 Oct 07, 2010 11:44 am
joannabanana wrote:Oh wow, that's really clever!

I was looking for a more formula-type answer, but this is way better. I doubt I'd be able to come up with something like this myself though, especially while writing the test. But I guess that's what these forums are for :)
Here's the formulaic (longer) approach.

P(HHHTT) = (1/2)^5 = 1/32.
We need to multiply by the number of ways to arrange HHHTT = 5!/3!*2! = 10.
10*1/32 = 10/32.

P(HHHHT) = (1/2)^5 = 1/32.
Since T could appear in any of the 5 positions, we multiply by 5.
5*1/32 = 5/32.

P(HHHHH) = (1/2)^5 = 1/32.

Since all of the above are good outcomes, we add the fractions:

10/32 + 5/32 + 1/32 = 16/32 = 1/2.
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
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Senior | Next Rank: 100 Posts
Posts: 36
Joined: Wed Aug 04, 2010 9:35 am
Thanked: 2 times

by aimscore » Thu Oct 07, 2010 1:37 pm
Hi Gmatguruny,

Pardon my ignorance but can u please explain how u reached this statement ?

Thus P(heads at least 3 times) = P(tails at least 3 times) = 1/2.
ie how did you get the value 1/2

Thanks
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
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 Oct 07, 2010 1:51 pm
aimscore wrote:Hi Gmatguruny,

Pardon my ignorance but can u please explain how u reached this statement ?

Thus P(heads at least 3 times) = P(tails at least 3 times) = 1/2.
ie how did you get the value 1/2

Thanks
When a coin is flipped 5 times, there are only 6 possible combinations of head and tails:

In the following 3 combinations, we get heads at least 3 times:
Heads 3 times, tails 2 times
Heads 4 times, tails 1 time
Heads 5 times, tails 0 times

In the following 3 combinations, we get tails at least 3 times.
Tails 3 times, heads 2 times
Tails 4 times, heads 1 time
Tails 5 times, heads 0 times

The probability of getting the first set of outcomes is the same as the probability of getting the second set of outcomes.

Thus, P(heads at least 3 times) = 1/2 and P(tails at least 3 times) = 1/2.

Does this help?
Last edited by GMATGuruNY on Thu Oct 07, 2010 7:21 pm, edited 1 time in total.
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
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Master | Next Rank: 500 Posts
Posts: 268
Joined: Wed Mar 17, 2010 2:32 am
Thanked: 17 times

by this_time_i_will » Thu Oct 07, 2010 6:33 pm
joannabanana wrote:Oh wow, that's really clever!

I was looking for a more formula-type answer, but this is way better. I doubt I'd be able to come up with something like this myself though, especially while writing the test. But I guess that's what these forums are for :)
Atleast 3 heads = heads 3 times OR heads 4 times OR heads 5 times

the total number of outcomes in 5 tosses = 2^5 = 32

heads 3 times = 5c3/32 = 10/32
heads 4 times = 5c4/32 = 5/32
heads 5 times = 5c5/32 = 1/32

adding we get 16/32 = 1/2
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Post new topic Post Reply
• Page 1 of 1