Made Est Test Series : How to find the expected number of dice rolls required to get 2 consecutive sixes? Can someone please explain an easy way to tackle such questions.

Question

Answer 1

Let the expected number of throws needed be $x$.

Now, if we throw a dice we can have 2 cases:

• The resulting throw is not a 6. So now the expected number of throws becomes $(x+1)$ as we are back where we started and we have used up 1 throw.
• The resulting throw is a 6. 
  • If the next throw also results in a 6 then expected number of throws is 2.
  • If the next throw is not a 6 then the expected number of throws becomes $(x+2)$ as we are back where we started and we have used up 2 throws.

$\therefore E[X]= \frac{5}{6}(x+1) + \frac{1}{36}(2) + \frac{5}{36}(x+2)$

$x= \frac{5}{6}(x+1) + \frac{1}{36}(2) + \frac{5}{36}(x+2)$

After solving you get $x=42$

 

Regarding how to solve these kinds of problems:

1. For above example you can have infinite amount of cases. So you can try above solution approach.
2. But if the number of cases are finite and you can find them easily then refer to this question GATE 2021 probability

Comments

This indeed helped. Thank you so much @rexritz