A) 100
B) 164
C) 170
D) 184
Correct Answer: B
Solution :
The probability of getting a double six in one throw with two dice \[=\frac{1}{6}\times \frac{1}{6}=\frac{1}{36}\] \[\therefore \] \[p=\frac{1}{36},q=1-p\] \[=1-\frac{1}{36}\] \[=\frac{35}{36}\] Now, \[{{\left( p+q \right)}^{m}}\] \[={{q}^{n}}{{+}^{n}}{{C}_{1}}{{q}^{n-1}}p{{+}^{n}}{{C}_{2}}{{q}^{n-2}}{{p}^{2}}\] \[+...{{+}^{n}}{{C}_{1}}{{q}^{n-r}}{{p}^{r}}+...+{{p}^{n}}\] The probability of getting atleast one double six in n throws with two dice \[={{\left( q+p \right)}^{n}}-{{q}^{n}}\] \[=1-{{q}^{n}}=1-{{\left( \frac{35}{36} \right)}^{n}}\] \[\therefore \] \[1-{{\left( \frac{35}{36} \right)}^{n}}>0.99\] \[\Rightarrow \] \[{{\left( \frac{35}{36} \right)}^{n}}<0.01\] \[\Rightarrow \] \[n\left( \log 35-\log 36 \right)<\log 0.01\] \[\Rightarrow \] \[n\left[ 15441-15536 \right]<-2\] \[\Rightarrow \] - 0.0122n < -2 \[\Rightarrow \] 0.0122 > 2 \[\Rightarrow \] \[n>\frac{2}{0.0122}\] \[\Rightarrow \] n > 163.9 So, the least value of n is 164.
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