Question

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to
1. 175/6⁵
2. 225/6⁵
3. 200/6⁵
4. 150/6⁵

Answer 1

Correct Answer - Option 1 : 175/6⁵

Since, the experiment should be end in the fifth throw of the die, so total number of outcomes are 6⁵.

Now, as the last two throws should be result in two fours

\(\frac{~}{\left( i \right)}\frac{~}{\left( ii \right)}\frac{4}{(\text{iii)}}\frac{4}{\left( iv \right)}\frac{4}{\left( v \right)}\) 

So, the third throw can be 1, 2, 3, 5 or 6 (not 4) Also, throw number (i) and (ii) cannot take two fours in succession, therefore number of possibilities for throw (i) and (ii) = 6² – 1 = 35

[When a pair of dice is thrown then (4, 4) occur only once]

Hence, the required probability is:

\(\Rightarrow \frac{5\times 35}{{{6}^{5}}}=\frac{175}{{{6}^{5}}}\)