If a die is rolled 30 times, there are 6^{30}

Arthur Pratt

Arthur Pratt

Answered question


If a die is rolled 30 times, there are 630 different sequences possible. The following question asks how many of these sequences satisfy certain conditions.
What fraction of these sequences have exactly three 4s and three 5s?

Answer & Explanation

Laura Worden

Laura Worden

Beginner2022-01-06Added 45 answers

Let 'S' be the sample space of rolling dice once.
Assuming a dice is fair, all the outcomes are equally likely.
When a dice is rolled once, there are 6 possible outcomes, when a dice is rolled twice, there are 62 possible outcomes, and, when the dice is rolled n times, there are 6n possible outcomes.
Here, the dice is rolled 30 times. Thus, the total number of possible outcomes are 630.
There are 30 places available in each sequence, the number of ways of selecting exactly three 4S=30C3
In the remaining 27 places, the number of ways of selecting exactly three 5S=27C3
In the remaining 24 places, any number other than 4 and 5 can take place.
The number of possible ways of filling the remaining 24 places =424
The total number of sequences with exactly three 4s and three 5s =(30C3)(27C3)(424)
Use the formula to compute the required probability:
Required fraction=Total number of sequences with exactly three 4s and three 5sTotal number of possible outcomes
Required fraction=(30C3)(27C3)(424)630

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