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If John flips a coin three times, what is the probability that he will

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If John flips a coin three times, what is the probability that he will  [#permalink]

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1. If John flips a coin three times, what is the probability that he will get a heads at least once?

2. If John flips a coin three times, what is the probability that it will turn up heads at least twice?
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Re: If John flips a coin three times, what is the probability that he will  [#permalink]

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New post 24 Apr 2010, 05:03
Actually, I want to know the concept, so can you kindly explain your answers in detail. There is no official answer as I just made these questions.
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Re: If John flips a coin three times, what is the probability that he will  [#permalink]

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New post 24 Apr 2010, 05:31
Hussain15 wrote:
1. If John flips a coin three times, what is the probability that he will get a heads at least once?

2. If John flips a coin three times, what is the probability that it will turn up heads at least twice?



1. If John flips a coin three times, what is the probability that he will get a heads at least once?

\(P(H\geq{1})=1-P(H=0)=1-(\frac{1}{2})^3=\frac{7}{8}\)

2. If John flips a coin three times, what is the probability that it will turn up heads at least twice?

\(P(H\geq{2})=P(H=2)+P(H=3)=\frac{3!}{2!}*(\frac{1}{2})^2*\frac{1}{2}+(\frac{1}{2})^3=\frac{1}{2}\)
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Re: If John flips a coin three times, what is the probability that he will  [#permalink]

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New post 21 Dec 2017, 18:38
Hi Bunuel,

I understand how to calculate P(H=0) is (\frac{1}{2})^3.
What does (\frac{1}{2}) + (\frac{1}{2}) + (\frac{1}{2}) mean ?

Could you please help me explain in detail how to calculate P(H=2) ?

Thanks in advance,
tanzzt

Bunuel wrote:
Hussain15 wrote:
1. If John flips a coin three times, what is the probability that he will get a heads at least once?

2. If John flips a coin three times, what is the probability that it will turn up heads at least twice?


1. If John flips a coin three times, what is the probability that he will get a heads at least once?

\(P(H\geq{1})=1-P(H=0)=1-(\frac{1}{2})^3=\frac{7}{8}\)

2. If John flips a coin three times, what is the probability that it will turn up heads at least twice?

\(P(H\geq{2})=P(H=2)+P(H=3)=\frac{3!}{2!}*(\frac{1}{2})^2*\frac{1}{2}+(\frac{1}{2})^3=\frac{1}{2}\)
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Re: If John flips a coin three times, what is the probability that he will  [#permalink]

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New post 21 Dec 2017, 20:04
1
tanzzt wrote:
Hi Bunuel,

I understand how to calculate P(H=0) is (\frac{1}{2})^3.
What does (\frac{1}{2}) + (\frac{1}{2}) + (\frac{1}{2}) mean ?

Could you please help me explain in detail how to calculate P(H=2) ?

Thanks in advance,
tanzzt

Bunuel wrote:
Hussain15 wrote:
1. If John flips a coin three times, what is the probability that he will get a heads at least once?

2. If John flips a coin three times, what is the probability that it will turn up heads at least twice?


1. If John flips a coin three times, what is the probability that he will get a heads at least once?

\(P(H\geq{1})=1-P(H=0)=1-(\frac{1}{2})^3=\frac{7}{8}\)

2. If John flips a coin three times, what is the probability that it will turn up heads at least twice?

\(P(H\geq{2})=P(H=2)+P(H=3)=\frac{3!}{2!}*(\frac{1}{2})^2*\frac{1}{2}+(\frac{1}{2})^3=\frac{1}{2}\)


The probability that the coin will turn up heads at least twice is the sum of the probability that it turn up heads twice and the probability that it turn up heads thrice: P(H=2) + P(H=3):

\(P(H=2) =\frac{3!}{2!}*(\frac{1}{2})^2*\frac{1}{2}\): Heads, Heads, Tails, can occur in several ways: HHT, HTH, THH, hence 3!/2! (the number of permutations of three letters HHT, where two H's are identical). Each of them has the equal probabilities: \((\frac{1}{2})^2*\frac{1}{2}=(\frac{1}{2})^3\)

\(P(H=3) = (\frac{1}{2})^3\).

21. Combinatorics/Counting Methods



For more:
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Hope it helps.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If John flips a coin three times, what is the probability that he will  [#permalink]

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New post 25 Dec 2017, 00:54
Hi Bunuel,

First of all, thanks for your reply.
Sorry that I'm still confused with the probability that it turns up heads exactly twice.
Does \frac{3!}{2!} mean flipping a coin three times & turning up exactly two heads?
So, if we have another question flipping a coin five times & turning up exactly three heads, it should be \frac{5!}{3!}. is that right?

Thanks.
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Re: If John flips a coin three times, what is the probability that he will  [#permalink]

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New post 25 Dec 2017, 01:06
1
tanzzt wrote:
Hi Bunuel,

First of all, thanks for your reply.
Sorry that I'm still confused with the probability that it turns up heads exactly twice.
Does \frac{3!}{2!} mean flipping a coin three times & turning up exactly two heads?
So, if we have another question flipping a coin five times & turning up exactly three heads, it should be \frac{5!}{3!}. is that right?

Thanks.


Two heads and one tail can occur in three ways: HHT, HTH, THH. So basically the number of ways this particular scenario can happen is the number of permutations of three letters HHT, where two H's are identical.

If it were 3 heads and 2 tails, then the number of ways this scenario can happen would be 5!/(3!2!) = 10:

HHHTT;
HHTHT;
HTHHT;
THHHT;
THHTH;
THTHH;
TTHHH;
HTTHH;
HHTTH;
HTHTH.

Again, this is the number of permutations of 5 letters HHHTT, out of which 3 H's and 2 T's are identical.

THEORY:

Permutations of \(n\) things of which \(P_1\) are alike of one kind, \(P_2\) are alike of second kind, \(P_3\) are alike of third kind ... \(P_r\) are alike of \(r_{th}\) kind such that: \(P_1+P_2+P_3+..+P_r=n\) is:

\(\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}\).

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is \(\frac{6!}{2!2!}\), as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be \(\frac{9!}{4!3!2!}\).

Hope it's clear.

P.S. For more follow the links in my previous post.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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Re: If John flips a coin three times, what is the probability that he will  [#permalink]

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New post 27 Dec 2017, 08:18
It was very clear Bunuel,
Many thanks for your explanation. You helped me a lot.
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Re: If John flips a coin three times, what is the probability that he will &nbs [#permalink] 27 Dec 2017, 08:18
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