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Senior Manager  Joined: 25 Oct 2008
Posts: 460
Location: Kolkata,India
The probability is 0.6 that an “unfair” coin will turn up tails on any  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 73% (01:31) correct 27% (01:39) wrong based on 335 sessions

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The probability is 0.6 that an “unfair” coin will turn up tails on any given toss. If the coin is tossed 3 times, what is the probability that at least one of the tosses will turn up tails?

A. 0.064
B. 0.36
C. 0.64
D. 0.784
E. 0.936

Where's my error:
P(1)+P(2)+P(3)
6/10.4/10.4/10 + 6/10.6/10.4/10 +6/10.6/10.6/10
=0.516 _________________
http://gmatclub.com/forum/countdown-beginshas-ended-85483-40.html#p649902

Originally posted by tejal777 on 02 Nov 2009, 05:51.
Last edited by Bunuel on 07 Feb 2019, 01:42, edited 2 times in total.
Edited the question and added the answer choices with OA
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
Re: The probability is 0.6 that an “unfair” coin will turn up tails on any  [#permalink]

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3
3
tejal777 wrote:
The probability is 0.6 that an “unfair” coin will turn up tails on any given toss. If the coin is tossed 3 times, what is the probability that at least one of the tosses will turn up tails?

Where's my error:
P(1)+P(2)+P(3)
6/10.4/10.4/10 + 6/10.6/10.4/10 +6/10.6/10.6/10
=0.516 The problem with your solution is that 1 or 2 tails can occur in several ways:

THH can occur in 3 ways: THH, HTH and HHT (basically it's # of arrangements of 3 letters THH out of whihc 2 H's are identical, which is 3!/2!=3). So, you should have P(THH)=0.6*0.4^2*3!/2!;
Similarly TTH can also occur in 3 way: TTH, THT and HTT. So, you should have P(TTH)=0.6^2*0.4*3!/2!.
Notice that we don't have the same issue with TTT, since it can occur only in one way: TTT.

$$P(T\geq{1})=0.6*0.4^2*\frac{3!}{2!}+0.6^2*0.4*\frac{3!}{2!}+0.6^3=0.936$$.

Though this question can be solved with an easier approach: $$P(T\geq{1})=1-P(H=3)=1-0.4^3=0.936$$ (the probability of at least one tails is 1 minus the probability of zero tails).

Hope it's clear.
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Manager  Joined: 29 Oct 2009
Posts: 176
GMAT 1: 750 Q50 V42 Re: The probability is 0.6 that an “unfair” coin will turn up tails on any  [#permalink]

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1
solve it this way:

probability that at least one of the tosses will turn up tails = 1 - probability that all will be heads

= 1 - (0.4*0.4*0.4)

= 1 - 0.064

= 0.936

where you went wrong:

probability that 1 is tails = (.6*.4*.4)+(.4*.6*.4)+(.4*.4*.6)

similarly for probability that 2 is tails.. consider all the cases..

in your solution, you have considered only one case each for both of the above..

even if you had solved it correctly, this method is longer and more complicated.. solve it as shown in my post above.. easier and quicker..

hope this helps!

cheers!
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Posts: 1353
Re: The probability is 0.6 that an “unfair” coin will turn up tails on any  [#permalink]

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I'm getting. Someone correct me if i'm wrong

First time:
probability of getting tails is .6 = .6

2nd time:
probability of getting heads is (.4)(.6) = .24

3rd time:
probability of getting tails is (.4)(.4)(.6) = .144

Probability of flipping unfair coin and getting tails is .984

Originally posted by lagomez on 02 Nov 2009, 06:05.
Last edited by lagomez on 02 Nov 2009, 06:17, edited 2 times in total.
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Re: The probability is 0.6 that an “unfair” coin will turn up tails on any  [#permalink]

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required probability = 1 - p(3 heads)

$$= 1 - (\frac{2}{5})^3 = \frac{117}{125}$$
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Re: The probability is 0.6 that an “unfair” coin will turn up tails on any  [#permalink]

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Wy we do not multiply 0.4^3by 3!/2! ?
So as a result we will have = 1- 0.4^3*3!/2!=0.808
I do understand that this is incorrect answer, but could not explain to myself what is wrong with it.
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
Re: The probability is 0.6 that an “unfair” coin will turn up tails on any  [#permalink]

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Van4ez wrote:
Wy we do not multiply 0.4^3by 3!/2! ?
So as a result we will have = 1- 0.4^3*3!/2!=0.808
I do understand that this is incorrect answer, but could not explain to myself what is wrong with it.

Factorial correction (multiplying by x!/(y!...)) account for different ways an event can occur. For example, THH can occur in 3 ways: THH, HTH and HHT, so we are multiplying by 3!/2! = 3 (3!/2! is the number of permutations of 3 letters THH where 2 T's are the same).

TTT case can occur only in one way Tails, Tails, Tails, so there is no need for factorial correction for this case.

Hope it's clear.

Theory on Combinations: math-combinatorics-87345.html

DS questions on Combinations: search.php?search_id=tag&tag_id=31
PS questions on Combinations: search.php?search_id=tag&tag_id=52

Tough and tricky questions on Combinations: hardest-area-questions-probability-and-combinations-101361.html

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Re: The probability is 0.6 that an “unfair” coin will turn up tails on any  [#permalink]

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_________________ Re: The probability is 0.6 that an “unfair” coin will turn up tails on any   [#permalink] 25 May 2019, 20:56
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