E(X): Find Probability of Rolling 4 Consecutive 6's with a Fair Dice

In summary, the conversation is discussing how to find the expected value of a random variable representing the number of times a fair six-sided dice needs to be rolled until 4 consecutive 6's are rolled. The answer, according to one participant, is 1554. Another participant is asking for clarification and input on the probability calculation, which involves calculating the probability of a 4-run of sixes using the binomial formula.
  • #1
undertoes
12
0
Let X be a random variable representing the number of times you need to roll (including the last roll) a fair six-sided dice until you get 4 consecutive 6's. Find E(X)?
answer is 1554.

I get confused with this, probability { X > n-5 }. I know that the last for throws must be 6's and the one before 'n-4 throws' must not be a 6. Any input please?
 
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  • #2
anyone?
 
  • #3
The probability that 4 of 4 throws are all sixes is q = 1/6^4. X is distributed Geometric with success probability q, and each "trial" represents four consecutive throws.

An alternative approach may be to calculate Prob{a 4-run of sixes} = 1 - Prob{not having a 4-run of sixes} using the binomial formula.
 

1. What is the concept of "E(X)" in probability?

The concept of "E(X)" is used to represent the expected value of a random variable X. It is the sum of the possible outcomes of X multiplied by their respective probabilities.

2. How is the probability of rolling 4 consecutive 6's with a fair dice calculated?

The probability of rolling 4 consecutive 6's with a fair dice can be calculated by taking the probability of rolling a 6 (1/6) and multiplying it by itself 4 times, since each roll is independent of the previous one. Therefore, the probability is (1/6)^4 = 1/1296.

3. Is the probability of rolling 4 consecutive 6's with a fair dice affected by previous rolls?

No, the probability of rolling 4 consecutive 6's with a fair dice is not affected by previous rolls. Each roll of the dice is independent and has the same probability of resulting in a 6.

4. Can the probability of rolling 4 consecutive 6's with a fair dice be greater than 1?

No, the probability of an event cannot be greater than 1. In this case, the probability of rolling 4 consecutive 6's is 1/1296, which is a very small probability.

5. How does the fairness of the dice affect the probability of rolling 4 consecutive 6's?

The fairness of the dice does not affect the probability of rolling 4 consecutive 6's. As long as the dice is fair, each roll has an equal probability of resulting in a 6.

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