• examPAL
    Most awarded test prep in the world
    Now free for 30 days

    Available with Beat the GMAT members only code

    MORE DETAILS
    examPAL
  • Kaplan Test Prep
    Free Practice Test & Review
    How would you score if you took the GMAT

    Available with Beat the GMAT members only code

    MORE DETAILS
    Kaplan Test Prep
  • Veritas Prep
    Free Veritas GMAT Class
    Experience Lesson 1 Live Free

    Available with Beat the GMAT members only code

    MORE DETAILS
    Veritas Prep
  • e-gmat Exclusive Offer
    Get 300+ Practice Questions
    25 Video lessons and 6 Webinars for FREE

    Available with Beat the GMAT members only code

    MORE DETAILS
    e-gmat Exclusive Offer
  • Magoosh
    Magoosh
    Study with Magoosh GMAT prep

    Available with Beat the GMAT members only code

    MORE DETAILS
    Magoosh
  • Varsity Tutors
    Award-winning private GMAT tutoring
    Register now and save up to $200

    Available with Beat the GMAT members only code

    MORE DETAILS
    Varsity Tutors
  • Target Test Prep
    5-Day Free Trial
    5-day free, full-access trial TTP Quant

    Available with Beat the GMAT members only code

    MORE DETAILS
    Target Test Prep
  • PrepScholar GMAT
    5 Day FREE Trial
    Study Smarter, Not Harder

    Available with Beat the GMAT members only code

    MORE DETAILS
    PrepScholar GMAT
  • EMPOWERgmat Slider
    1 Hour Free
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    EMPOWERgmat Slider
  • Economist Test Prep
    Free Trial & Practice Exam
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    Economist Test Prep

If six coins are flipped simultaneously, the probability ...

This topic has 5 expert replies and 2 member replies
gmattesttaker2 Legendary Member Default Avatar
Joined
14 Feb 2012
Posted:
641 messages
Followed by:
8 members
Upvotes:
11

If six coins are flipped simultaneously, the probability ...

Post Sun Jun 15, 2014 10:02 pm
Hello,

Can you please tell me if my approach is correct here? I was not clear with the official explanation.

If six coins are flipped simultaneously, the probability of getting at least one heads and at least one tails is closest to:

A) 3%
B) 6%
C) 75%
D) 94%
E) 97%

OA: 97%

P( At least one heads and at least one tails ) = P (At least 1 head). P(At least 1 tail)


P( At least 1 head ) = 1 - P( No heads ) = 1 - (1/2)^6 = 1 - (1/64) = 63/64

Similarly, P ( At least 1 tail ) = 63/64

Hence, P( At least one heads and at least one tails ) = P (At least 1 head). P(At least 1 tail)
= 63/64 x 63/64
= approx. 1

Hence, 97%

I was just wondering if this is correct? Thanks for your help.

Regards,
Sri

  • +1 Upvote Post
  • Quote
  • Flag

GMAT/MBA Expert

Post Thu Jul 28, 2016 10:35 am
Anantjit wrote:
Hi Brent,

How did AND become OR in complement.

So is it everytime while we take a complement of AND CONDITION it becomes OR condition?
Rather than try to create rules for each situation, I think it's better to understand what the complement means in each case.

For example, if a question asks us to find P(at least one head) then we can say:
P(at least one head) = 1 - P(NOT at least one head)
= 1 - P(zero heads)
= 1 - P(all tails)

Another example: if a question asks us to find P(at least two heads) then we can say:
P(at least two heads) = 1 - P(NOT at least two heads)
= 1 - P(1 head OR 0 heads)

Does that help?

Cheers,
Brent

_________________
Brent Hanneson – Founder of GMATPrepNow.com
Use our video course along with Beat The GMAT's free 60-Day Study Guide

Check out the online reviews of our course
Come see all of our free resources

  • +1 Upvote Post
  • Quote
  • Flag
GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!

GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
Joined
12 Sep 2012
Posted:
2640 messages
Followed by:
113 members
Upvotes:
625
Target GMAT Score:
V51
GMAT Score:
780
Post Thu Aug 04, 2016 9:09 pm
Anantjit wrote:
How did AND become OR in complement.

So is it everytime while we take a complement of AND CONDITION it becomes OR condition?
x AND y is actually a subset of x OR y, at least if we're using OR in the logical sense, where OR = "at least one of ..."

For instance, suppose I'm thinking about cars and trucks, some of which are fast and some of which are big. Cars that are fast AND big are a subset of cars that are fast OR big, since they appear in the fast set and in the big set.

  • +1 Upvote Post
  • Quote
  • Flag
Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now!

GMAT/MBA Expert

Post Mon Jun 16, 2014 3:26 am
gmattesttaker2 wrote:
If six coins are flipped simultaneously, the probability of getting at least one heads and at least one tails is closest to:

A) 3%
B) 6%
C) 75%
D) 94%
E) 97%

OA: 97%
P(good outcome) = 1 - P(bad outcome).
Here, a GOOD outcome is to to get at least one heads and at least one tails.
Thus, a BAD outcome is to NOT get at least one heads and at least one tails -- in other words, to get the SAME result on all 6 flips (all heads or all tails).
Thus:
P(at least one heads and at least one tails) = 1 - P(all 6 flips are the same).

P(all 6 flips are the same):
The first flip can be heads or tails.
P(the 2nd flip is the same as the first) = 1/2.
P(the 3rd flip is the same as the preceding flips) = 1/2.
P(the 4th flip is the same as the preceding flips) = 1/2.
P(the 5th flip is the same as the preceding flips) = 1/2.
P(the 6th flip is the same as the preceding flips) = 1/2.
To combine these probabilities, we multiply:
1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/32 = 3/96 ≈ 0.03.

P(at least one heads and at least one tails):
1 - 0.03 = 0.97 = 97%.

The correct answer is E.

Quote:
P( At least 1 head ) = 1 - P( No heads ) = 1 - (1/2)^6 = 1 - (1/64) = 63/64

Similarly, P ( At least 1 tail ) = 63/64
There is quite a bit of overlap between the two probabilities calculated above.

P(at least 1 heads) is composed of the following outcomes:
1H, 5T
2H, 4T
3H, 3T
4H, 2T
5H, 1T

6H.

P(at least 1 tails) is composed of the following outcomes:
1T, 5H
2T, 4H
3T, 3H
4T, 2H
5T, 1H

6T.

In your solution, all of the probabilities in red are counted twice.

_________________
Mitch Hunt
GMAT Private Tutor
GMATGuruNY@gmail.com
If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.
Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.

  • +1 Upvote Post
  • Quote
  • Flag
Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.

GMAT/MBA Expert

Post Mon Jun 16, 2014 6:49 am
gmattesttaker2 wrote:
If six coins are flipped simultaneously, the probability of getting at least one heads and at least one tails is closest to:

A) 3%
B) 6%
C) 75%
D) 94%
E) 97%

A slightly different approach (with the same answer Smile)

When it comes to probability questions involving "at least," it's best to try using the complement.
That is, P(Event A happening) = 1 - P(Event A not happening)
So, here we get: P(getting at least one heads and at least one tails) = 1 - P(not getting at least one heads and at least one tails)

What does it mean to not get at least one heads and at least one tails? It means getting EITHER zero heads OR zero tails.

So, we can write: P(getting at least one heads and at least one tails) = 1 - P(getting zero heads OR zero tails)
= 1 - P(getting ALL tails OR ALL heads)

P(getting ALL tails OR ALL heads)
= (tails on 1st toss AND tails on 2nd toss AND tails on 3rd toss AND tails on 4th toss AND tails on 5th toss AND tails on 6th toss OR heads on 1st toss AND heads on 2nd toss AND heads on 3rd toss AND heads on 4th toss AND heads on 5th toss AND heads on 6th toss)
= [1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2] + [1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2]
= [1/64] + [1/64]
= 2/64
= 1/32

So, P(getting at least one heads and at least one tails) = 1 - 1/32
= 31/32
97%
= A

Cheers,
Brent

_________________
Brent Hanneson – Founder of GMATPrepNow.com
Use our video course along with Beat The GMAT's free 60-Day Study Guide

Check out the online reviews of our course
Come see all of our free resources

  • +1 Upvote Post
  • Quote
  • Flag
GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!
prachi18oct Master | Next Rank: 500 Posts
Joined
27 Apr 2014
Posted:
269 messages
Followed by:
5 members
Upvotes:
8
Post Mon Jun 16, 2014 9:59 pm
Can I use the below approach as well??

P(Atleast 1 head and atleast 1 tail) => 1-P(All heads or All tails)
=> only when HHHHHH or TTTTTT comes then we will not get atleast one head and one tail
=>Total ways of choosing 6 objects from set of 12 = 12C6 P(HHHHHH or TTTTTT) = 924
=> Total ways of getting HHHHHH or TTTTTT =>2
=>1-2/924
However I am getting almost 99%
Please explain where I did it wrong?Is this approach not correct?

I think I have taken incorrectly the total ways of selecting from 12 objects..Is it because they are similar?12C6 will have similar selections also .How can I correct this?I have to remove the similar selections but how?

  • +1 Upvote Post
  • Quote
  • Flag

GMAT/MBA Expert

Post Mon Jun 16, 2014 10:15 pm
Hi prachi18oct,

Yes, you can use the approach that you described.

You have made a math error though. Since there are 6 coins and each coin will either be a "heads" or a "tails", there are NOT 12 items to choose from. The possible number of outcomes is 2^6 = 64.

You are correct that there are only 2 ways of NOT getting what you "want" (at least 1 head and at least 1 tail).

Thus, there are 2/64 of NOT getting what you want and 1 - 2/64 = 62/64 = 31/32 ways of getting what you want.

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

_________________
Contact Rich at Rich.C@empowergmat.com

  • +1 Upvote Post
  • Quote
  • Flag
Anantjit Junior | Next Rank: 30 Posts Default Avatar
Joined
13 Aug 2015
Posted:
12 messages
Post Thu Jul 28, 2016 9:51 am
Brent@GMATPrepNow wrote:
gmattesttaker2 wrote:
If six coins are flipped simultaneously, the probability of getting at least one heads and at least one tails is closest to:

A) 3%
B) 6%
C) 75%
D) 94%
E) 97%

A slightly different approach (with the same answer Smile)

When it comes to probability questions involving "at least," it's best to try using the complement.
That is, P(Event A happening) = 1 - P(Event A not happening)
So, here we get: P(getting at least one heads and at least one tails) = 1 - P(not getting at least one heads and at least one tails)

What does it mean to not get at least one heads and at least one tails? It means getting EITHER zero heads OR zero tails.

So, we can write: P(getting at least one heads and at least one tails) = 1 - P(getting zero heads OR zero tails)
= 1 - P(getting ALL tails OR ALL heads)

P(getting ALL tails OR ALL heads)
= (tails on 1st toss AND tails on 2nd toss AND tails on 3rd toss AND tails on 4th toss AND tails on 5th toss AND tails on 6th toss OR heads on 1st toss AND heads on 2nd toss AND heads on 3rd toss AND heads on 4th toss AND heads on 5th toss AND heads on 6th toss)
= [1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2] + [1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2]
= [1/64] + [1/64]
= 2/64
= 1/32

So, P(getting at least one heads and at least one tails) = 1 - 1/32
= 31/32
97%
= A

Cheers,
Brent
Hi Brent,

How did AND become OR in complement.

So is it everytime while we take a complement of AND CONDITION it becomes OR condition?

  • +1 Upvote Post
  • Quote
  • Flag

Best Conversation Starters

1 lheiannie07 112 topics
2 ardz24 71 topics
3 Roland2rule 69 topics
4 LUANDATO 53 topics
5 swerve 45 topics
See More Top Beat The GMAT Members...

Most Active Experts

1 image description GMATGuruNY

The Princeton Review Teacher

154 posts
2 image description Rich.C@EMPOWERgma...

EMPOWERgmat

107 posts
3 image description Jeff@TargetTestPrep

Target Test Prep

106 posts
4 image description Scott@TargetTestPrep

Target Test Prep

98 posts
5 image description EconomistGMATTutor

The Economist GMAT Tutor

91 posts
See More Top Beat The GMAT Experts