I roll a die repeatedly until I get 6, and then count the number of 3s I got. What’s my expected number of 3s?

Consider the following experiment. I roll a die repeatedly until the die returns 6, then I count the number of times 3 appeared in the random variable X. What is E[X]?

Thoughts: I expect to roll the die 6 times before 6 appears (this part is geometric), and on the preceding 5 rolls each roll has a 1/5 chance of returning a 3. Treating this as binomial, I therefore expect to count 3 once, so E[X]=1.

Problem: Don’t know how to model this problem mathematically. Hints would be appreciated.

Answer

We can restrict ourselves to dice throws with outcomes 3 and 6. Among these throws, both outcomes are equally likely. This means that the index Y of the first 6 is geometrically distributed with parameter 12, hence E(Y)=2. The number of 3s occuring before the first 6 equals Y1 and has expected value 1.

Attribution
Source : Link , Question Author : nettle , Answer Author : user133281

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