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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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24 Apr 2010, 00:28
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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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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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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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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 ?

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?

$$P(H\geq{2})=P(H=2)+P(H=3)=\frac{3!}{2!}*(\frac{1}{2})^2*\frac{1}{2}+(\frac{1}{2})^3=\frac{1}{2}$$
Math Expert
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Posts: 47019
Re: If John flips a coin three times, what is the probability that he will [#permalink]

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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 ?

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:
ALL YOU NEED FOR QUANT ! ! !

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

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25 Dec 2017, 00:54
Hi Bunuel,

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.
Math Expert
Joined: 02 Sep 2009
Posts: 47019
Re: If John flips a coin three times, what is the probability that he will [#permalink]

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

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.

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

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