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

 It is currently 14 Dec 2019, 17:05

### 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

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

# If a fair two-sided coin is flipped 6 times, what is the probability

Author Message
TAGS:

### Hide Tags

Senior Manager
Status: Gathering chakra
Joined: 05 Feb 2018
Posts: 444
If a fair two-sided coin is flipped 6 times, what is the probability  [#permalink]

### Show Tags

06 Mar 2019, 14:05
3
00:00

Difficulty:

65% (hard)

Question Stats:

58% (02:29) correct 42% (02:57) wrong based on 95 sessions

### HideShow timer Statistics

If a fair two-sided coin is flipped 6 times, what is the probability that tails is the result at least twice but at most 5 times?

A) $$\frac{5}{8}$$

B) $$\frac{3}{4}$$

C) $$\frac{7}{8}$$

D) $$\frac{57}{64}$$

E) $$\frac{15}{16}$$
Intern
Joined: 15 Jan 2019
Posts: 45
Location: Pakistan
Concentration: Marketing, Human Resources
GPA: 3.12
WE: Other (Other)
Re: If a fair two-sided coin is flipped 6 times, what is the probability  [#permalink]

### Show Tags

06 Mar 2019, 21:38
The trick I am using to solve this question was taught to me by a channel on YouTube named Salman Gaffar. It’s Probability 07

So first let’s list down the possible outcomes

1) H H H H H H
2) H H H H H T
3) H H H H T T
4) H H H T T T
5) H H T T T T
6) H T T T T T
7) T T T T T T

We want to find out the probability of Atleast 2 tails and atmost 5 tails so option 1,2,7 are unfavourable and 3,4,5,6 are favourable

We can find probability of either one of them. Remember if we find probability of unfavourable outcomes we need to subtract it by 1 to get the probability of favourable outcomes. Let me show how

Let’s find the probability of option 1

1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/64

For option 2

1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/64 x 6!/5! = 6/64

For option 7

1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/64

So it’s either option 1 OR option 2 OR option 7 as unfavourable outcome

Since it’s OR we add, if its AND we multiply

1/64 + 1/64 + 6/64 = 8/64 or 1/8

1 - 1/8 = 7/8

Option C is the right answer

Posted from my mobile device
Intern
Joined: 11 May 2018
Posts: 24
Location: India
Re: If a fair two-sided coin is flipped 6 times, what is the probability  [#permalink]

### Show Tags

07 Mar 2019, 09:30
1
for getting at least two tails or at most 5 tails:
1) Probablity of getting 2 tails = 6C2/64 : (6C2 --> two tails can be from any of the 6 flips)
2) Probablity of getting 3 tails = 6C3/64
3) probablity of getting 4 tails = 6C4/64
4) probablity of getting 5 tails = 6C5/64

Probablity of at least 2 tails but at most 5 tails =(6C2+6C3+6C4+6C5)/64 = 56/64 = 7/8
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8701
Location: United States (CA)
Re: If a fair two-sided coin is flipped 6 times, what is the probability  [#permalink]

### Show Tags

10 Mar 2019, 19:45
1
energetics wrote:
If a fair two-sided coin is flipped 6 times, what is the probability that tails is the result at least twice but at most 5 times?

A) $$\frac{5}{8}$$

B) $$\frac{3}{4}$$

C) $$\frac{7}{8}$$

D) $$\frac{57}{64}$$

E) $$\frac{15}{16}$$

We need to determine the following probabilities:

TTHHHH, TTTHHH, TTTTHH, and TTTTTH

P(TTHHHH) = (1/2)^6 = 1/64

TTHHHH can be arranged in 6C2 = 6!(2! x 4!) = (6x5)/2 = 15 ways, so the probability is 15/64.

P(TTTHHH) = (1/2)^6 = 1/64

TTTHHH can be arranged in 6C3 = 6!(3! x 3!) = (6x5x4)/3! = 20 ways, so the probability is 20/64.

P(TTTTHH) = (1/2)^6 = 1/64

TTTTHH can be arranged in 6C4 = 6!(2! x 4!) = (6x5)/2 = 15 ways, so the probability is 15/64.

P(TTTTTH) = (1/2)^6 = 1/64

TTTTTH can be arranged in 6C5 = 6!/5! = 6 ways, so the probability is 6/64.

Thus, the total probability is (15 + 20 + 15 + 6)/64 = 56/64 = 7/8.

_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Director
Joined: 24 Nov 2016
Posts: 972
Location: United States
If a fair two-sided coin is flipped 6 times, what is the probability  [#permalink]

### Show Tags

21 Aug 2019, 04:47
energetics wrote:
If a fair two-sided coin is flipped 6 times, what is the probability that tails is the result at least twice but at most 5 times?

A) $$\frac{5}{8}$$
B) $$\frac{3}{4}$$
C) $$\frac{7}{8}$$
D) $$\frac{57}{64}$$
E) $$\frac{15}{16}$$

METHOD 1 (find the probability of each case and add them together)
TTHHHH: 1/64*(6!/2!4!)=1/64*15=15/64
TTTHHH: 1/64*(6!/3!3!)=1/64*20=20/64
TTTTHH: 1/64*(6!/4!2!)=1/64*15=15/64
TTTTTH: 1/64*(6!/5!1!)=1/64*6=6/64
total: 56/64=28/32=14/16=7/8

METHOD 2 (find the probability of not getting the desired result and subtract 1 from the result)
HHHHHH [no tails]: 1/64*(6!/6!)=1/64*1=1/64
TTTTTTT [all tails]: 1/64*(6!/6!)=1/64*1=1/64
THHHHH [one tails]: 1/64*(6!/1!5!)=1/64*6=6/64
total: 1-(8/64)=56/64=28/32=14/16=7/8

Math Expert
Joined: 02 Sep 2009
Posts: 59725
Re: If a fair two-sided coin is flipped 6 times, what is the probability  [#permalink]

### Show Tags

21 Aug 2019, 05:02
energetics wrote:
If a fair two-sided coin is flipped 6 times, what is the probability that tails is the result at least twice but at most 5 times?

A) $$\frac{5}{8}$$

B) $$\frac{3}{4}$$

C) $$\frac{7}{8}$$

D) $$\frac{57}{64}$$

E) $$\frac{15}{16}$$

The opposite even would be 0, 1, or 6 tails.

$$P(t = 0) = P(h = 6) = (\frac{1}{2})^6$$

$$P(t = 1) = \frac{6!}{5!}*(\frac{1}{2})*(\frac{1}{2})^5$$. We are multiplying by 6!/5! because thhhhh can occur in 6!/5! = 6 different ways: thhhhh, hthhhh, hhthhh, hhhthh, hhhhth, hhhhht.

$$P(t = 6) = (\frac{1}{2})^6$$

$$P (t = 2, 3, 4, \ or \ 5) = 1- P(t = 0, 1, \ or \ 6) = 1 - ((\frac{1}{2})^6 + \frac{6!}{5!}*(\frac{1}{2})*(\frac{1}{2})^5 + (\frac{1}{2})^6) = \frac{7}{8}$$

_________________
Re: If a fair two-sided coin is flipped 6 times, what is the probability   [#permalink] 21 Aug 2019, 05:02
Display posts from previous: Sort by