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

Question

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

Bunuel · Accepted Answer

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 .

Answer: B.

Cornelius · Answer

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

ezhilkumarank · Answer

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%.

Hence answer is 2 (B).

sagarsir · Answer

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

Kris01 · Answer

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

sri30kanth · Answer

Bunuel,

Could you please help us with the solution for this problem? Thanks

varun325 · Answer

My approach:

For probability of 1( NO. 4 will come) Min no. of throws =6
For probability of 0.25 ( no. 4 will come), Min No. of throws= 6*0.25=1.5
so minimum no. of throws required are 2.

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

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