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# How many times must a fair six sided die be rolled so that

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How many times must a fair six sided die be rolled so that [#permalink]  01 Sep 2010, 08:23
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How many times must a fair six sided die be rolled so that there is at least 1/4 probability that a 4 is rolled ?

A. 1
B. 2
C. 3
D. 4
E. 5
[Reveal] Spoiler: OA

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Re: How many times must a fair six sided die be rolled so that [#permalink]  01 Sep 2010, 21:58
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My approach:

 How many times must a fair six sided die be rolled so that there is at least 1/4 probability that a 4 is rolled?

Probability of at least 1/4 that the number 4 is rolled is same as the probability of the at most 3/4 that the number other than 4 is rolled.

First roll -- probability of not getting a number 4 -- 5/6 ~ 16.67 * 5 > 80%. Hence the probability of getting a 4 is less than 20%.

Second roll -- probability of not getting a number 4 -- 5/6 * 5/6 -- 25/36 ~< 75 %. Hence the probability of getting a 4 is approx. greater than 25%.

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Re: How many times must a fair six sided die be rolled so that [#permalink]  22 Aug 2014, 10:06
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sri30kanth wrote:
Bunuel,

How many times must a fair six sided die be rolled so that there is at least 1/4 probability that a 4 is rolled ?

A. 1
B. 2
C. 3
D. 4
E. 5

The probability of NOT getting a 4 when rolling once is 5/6.
The probability of getting a 4 when rolling once is 1 - 5/6.

The probability of NOT getting a 4 when rolling twice is 5/6*5/6 = (5/6)^2.
The probability of getting a 4 when rolling twice is 1 - (5/6)^2.

The probability of NOT getting a 4 when rolling n times is (5/6)^n.
The probability of getting a 4 when rolling n times is 1 - (5/6)^n.

So, we need such n for which $$1 - (\frac{5}{6})^n \geq{ \frac{1}{4}}$$ --> $$(\frac{5}{6})^n \leq{ \frac{3}{4}}$$ --> $$n_{min}=2$$.

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Re: How many times must a fair six sided die be rolled so that [#permalink]  01 Sep 2010, 10:01
Let n be the number of times the dice is rolled to ensure that the probability of a 4 is at least 1/4.

Thus,
n*(probability of rolling a 4 in one turn)>= 1/4
n*(1/6)>= 1/4
n>6/4---->n>= 1.5
Hence "minimum number of times to ensure that the probability that a 4 is rolled is at least 1/4" >= 1.5 = 2 (since we cannot have 1.5 rolls)

Ans B
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Re: How many times must a fair six sided die be rolled so that [#permalink]  04 Feb 2013, 02:34
Cornelius wrote:
Let n be the number of times the dice is rolled to ensure that the probability of a 4 is at least 1/4.

Thus,
n*(probability of rolling a 4 in one turn)>= 1/4
n*(1/6)>= 1/4
n>6/4---->n>= 1.5
Hence "minimum number of times to ensure that the probability that a 4 is rolled is at least 1/4" >= 1.5 = 2 (since we cannot have 1.5 rolls)

Ans B

By this logic, in 6 throws, you have a probability of getting 4 as 1. I doubt if your approach is correct.
And in 7 and above throws the probability to get 4 will be more than 1. I dont know if I am missing something
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Re: How many times must a fair six sided die be rolled so that [#permalink]  16 Feb 2013, 15:27
How I look at it is that since the number of throws is lesser, there is lesser chance of all numbers coming up. Hence, 4 has a chance of coming up more than 1/4 times in two throws.

Another way to look at it is that in two throws only two numbers can come up and the chance of 4 coming up once in two trials will be higher than 1/6

4 not 4
not 4 4

1/6*5/6+5/6*1/6 = 25/36>1/4

sagarsingh wrote:
Cornelius wrote:
Let n be the number of times the dice is rolled to ensure that the probability of a 4 is at least 1/4.

Thus,
n*(probability of rolling a 4 in one turn)>= 1/4
n*(1/6)>= 1/4
n>6/4---->n>= 1.5
Hence "minimum number of times to ensure that the probability that a 4 is rolled is at least 1/4" >= 1.5 = 2 (since we cannot have 1.5 rolls)

Ans B

By this logic, in 6 throws, you have a probability of getting 4 as 1. I doubt if your approach is correct.
And in 7 and above throws the probability to get 4 will be more than 1. I dont know if I am missing something

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Re: How many times must a fair six sided die be rolled so that [#permalink]  22 Aug 2014, 08:10
Bunuel,

Re: How many times must a fair six sided die be rolled so that   [#permalink] 22 Aug 2014, 08:10
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