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# A fair 2-sided coin is flipped 6 times. What is the

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A fair 2-sided coin is flipped 6 times. What is the [#permalink]  16 Dec 2009, 09:01
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A fair 2-sided coin is flipped 6 times. What is the probability that tails will be the result at least twice, but not more than 5 times?

(A) 5/8
(B) 3/4
(C) 7/8
(D) 57/64
(E) 15/16
[Reveal] Spoiler: OA

Last edited by Bunuel on 10 Feb 2012, 11:51, edited 1 time in total.
Edited the question and added the OA
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Re: Probability [#permalink]  16 Dec 2009, 09:16
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melissawlim wrote:
A fair 2-sided coin is flipped 6 times. What is the probability that tails will be the result at least twice, but not more than 5 times?

(A) 5/8
(B) 3/4
(C) 7/8
(D) 57/64
(E) 15/16

It would be easier to calculate the probability of opposite event and subtract it from 1.
Opposite event: 0 tail, 1 tail, 6 tails.

Probability of getting no tails: $$\frac{1}{2^6}=\frac{1}{64}$$;

Probability of getting 1 tail: $$6C1*\frac{1}{2^6}=\frac{6}{64}$$, we must multiply by 6C1 or by 6 as tail can occur for any flip from 6, hence in 6 ways (*see more about this case below);

Probability of getting 6 tails: $$\frac{1}{2^6}=\frac{1}{64}$$

$$P=1-(\frac{1}{64}+\frac{6}{64}+\frac{1}{64})=\frac{56}{64}=\frac{7}{8}$$

* Counting probability of getting 1 tail when flipped 6 times: 1 tail can occur in 6 different ways:

THHHHH
HTHHHH
HHTHHH
HHHTHH
HHHHTH
HHHHHT

Generally probability of occurring event k times in n-time sequence could be expressed as:

$$P = C^n_k*p^k*(1-p)^{n-k}$$

In our case $$k=1$$ and $$n=6$$, so we get:

$$P = C^6_1*\frac{1}{2}*\frac{1}{2^5}=6*\frac{1}{2^6}=\frac{6}{64}$$

So $$\frac{1}{64}$$ should be multiplied by $$C^6_1$$, which is $$6$$.
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Re: Probability [#permalink]  16 Dec 2009, 09:23
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lets find it out the other way
P of all head(or no tail)=1/2^6=1/64
P of all tail (or no head)=1/2^6=1/64
P of one tail=1/2^6 * 6 { multiply by 6 as there are 6 ways we can get one tail and each is having a probability of 1/2}
tot=1/64 + 1/64 + 6/64=1/8
reqd P=1-tot=1-1/8=7/8

C
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Re: A fair 2-sided coin is flipped 6 times. What is the [#permalink]  19 Jun 2014, 19:11
Hi Bunuel,

Shouldn't the value of K be equal to 2 as we are looking for atleast twice.And secondly can you please explain me how do we restrict here that tails do not occur for more than five times by using the formula.
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Re: A fair 2-sided coin is flipped 6 times. What is the [#permalink]  20 Jun 2014, 04:18
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maggie27 wrote:
Hi Bunuel,

Shouldn't the value of K be equal to 2 as we are looking for atleast twice.And secondly can you please explain me how do we restrict here that tails do not occur for more than five times by using the formula.

The example there with formula is for getting 1 tail when flipped 6 times.

The solution uses P(good) = 1 - P(bad). So, P(2, 3, 4, or 5 tails) = 1 - P(0, 1, or 6 tails.)
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Re: A fair 2-sided coin is flipped 6 times. What is the [#permalink]  20 Jun 2014, 06:29
Thank You Bunuel
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Re: A fair 2-sided coin is flipped 6 times. What is the [#permalink]  20 Jun 2014, 07:12
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melissawlim wrote:
A fair 2-sided coin is flipped 6 times. What is the probability that tails will be the result at least twice, but not more than 5 times?

(A) 5/8
(B) 3/4
(C) 7/8
(D) 57/64
(E) 15/16

Total possible outcomes when coined is tossed 6 time=2^6=4*4*4=64
Total possible outcomes getting 2 or 3 or 4 or 5 tails= 6C2+6C3+6C4+6C5=(6*5)/2+(6*5*4)/(3*2)+(6*5)/2+6=15+20+15+6=56
Probability of getting atleast 2 but not more than 5 times tails=56/64=7/8

Ans=C
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Re: A fair 2-sided coin is flipped 6 times. What is the [#permalink]  20 Jun 2014, 22:35
Forgot considering 0 events))
Re: A fair 2-sided coin is flipped 6 times. What is the   [#permalink] 20 Jun 2014, 22:35
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