If six coins are flipped simultaneously, the probability of

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Tue Sep 18, 2018 4:58 pm

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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 E

Source: Veritas Prep

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Tue Sep 18, 2018 9:10 pm

BTGmoderatorDC 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 E

Source: Veritas Prep

Instead of calculating the probability of getting at least one head and at least one tail, let's calculate the probability of getting NO head and the probability of getting NO tail. We'll then subtract these values from 1 to get the answer.

Probability of getting NO head = Probability of getting ALL tail = (1/2)^6

Similarly,

Probability of getting NO tail = Probability of getting ALL head = (1/2)^6

Thus, the probability of getting at least one head and at least one tail = 1 - (1/2)^6 - (1/2)^6 = 1 - (1/2)^5 = 1 - 1/32 = 31/32 = ~97%

The correct answer: E

Hope this helps!

-Jay
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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Wed Sep 19, 2018 6:03 am

BTGmoderatorDC 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 E

Source: Veritas Prep

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
â‰ˆ 97%

Answer: E

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Sun Sep 23, 2018 12:59 pm

BTGmoderatorDC 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%
Source: Veritas Prep

\[?\,\,\, = \,\,\,1 - P\left( {{\text{in}}\,\,6\,\,{\text{flips}},\,\,6H\,\,{\text{or}}\,\,6T} \right)\,\,\, = \,\,\,1 - {?_{{\text{temporary}}}}\]
\[\left. \begin{gathered}
{\text{Total}} = \,\,{2^6}\,\,\,{\text{equiprobables}} \hfill \\
{\text{Favorable}} = \,2\,\,\,\,\left( {6H\,\,{\text{or}}\,\,6T} \right)\,\, \hfill \\
\end{gathered} \right\}\,\,\,\, \Rightarrow \,\,\,\,{?_{{\text{temporary}}}} = \frac{2}{{{2^6}}} = \frac{1}{{32}} = \frac{{96 + 4}}{{32}}\% = 3\frac{1}{8}\% \]
\[?\,\,\, \cong \,\,\,100\% - 3\% \]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8136; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Sun Sep 23, 2018 3:42 pm

BTGmoderatorDC 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%

There are 2^6 = 64 ways the six coins could be flipped. Of the these 64 ways, only two of them (all heads and all tails) do not have at least one head or at least one tail. Therefore, the probability of getting at least one head and at least one tail is 62/64 = 0.96875 â‰ˆ 97%.

Answer: E

Scott Woodbury-Stewart
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[email protected]

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[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] » Sun Sep 23, 2018 7:32 pm

Hi All,

We're told that six coins are flipped simultaneously. We're asked which probability is closest to getting AT LEAST one heads and AT LEAST one tails. This question comes down to a bit of probability math (which is essentially just about doing the necessary Arithmetic).

With 6 coins, there are (2)(2)(2)(2)(2)(2) = 64 possible outcomes. The ONLY outcomes that would not include at least one head AND at least one tail would be 'all heads' or 'all tails' --> HHHHHH or TTTTTT. Those are 2 of the 64 possible outcomes, meaning that 2/64 = 1/32 outcomes don't fit what we're looking for.

32 divides into 100 a little more than 3 times, so 1/32 = about .03 = about 3%. Thus, the other 97% of outcomes gives us what we're looking for.

Final Answer: E

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

Contact Rich at [email protected]

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