question_answer

How many dice must be thrown so that there is a batter than even chance of obtaining a six?

Answer:

Let n dice be thrown and let X denotes the number of sixes. Then.  \[P(X=r){{=}^{n}}{{C}_{r}}{{\left( \frac{1}{6} \right)}^{r}}{{\left( \frac{5}{6} \right)}^{r}};r=0,\,\,1,\,\,2...,n\] We have to find the smallest value of n for which P(X = 0) is less than \[\frac{1}{2}.\] Now, \[P(X=0)<\frac{1}{2}\] \[\Rightarrow \] \[{{\left( \frac{5}{6} \right)}^{n}}<\frac{1}{2}\] Clearly, \[\frac{5}{6}<|\frac{1}{2},\,\,{{\left( \frac{5}{6} \right)}^{2}}<|\frac{1}{2},{{\left( \frac{5}{6} \right)}^{3}}<|\frac{1}{2},\] But \[{{\left( \frac{5}{6} \right)}^{4}}=\frac{625}{1296}<\frac{1}{2}\] \[\therefore \] \[P(X=0)<\frac{1}{2}\] \[\Rightarrow \] \[{{\left( \frac{5}{6} \right)}^{n}}<\frac{1}{2}\] \[\Rightarrow \] n = 4, 5? Thus, at least 4 dice must be thrown.