A fair die is rolled 1,000 times. What is the probability of rolling the same number 5 times in a row?

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A fair die is rolled 1,000 times. What is the probability of rolling the same number 5 times in a row? How do you solve this type of question for variable number of throws and number of repeats?

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How to calculate the probability of rolling 6 at least 5 times in a row, out of 50 tries?

This question is very similar to yours. You have to modify it for not having exactly the number 6 but the same number, so you will have different probabilities. Try calculating it for one specific number and then expand your approach.