three dice are rolled simultaneously until all three show th

three dice are rolled simultaneously until all three show the same number. what is the probability that they are rolled more than 10 times? given they are rolled more than 10 times what is the probability that they are rolled more than 20 times?

Solution

a)

There are 6*6*6 = 216 possible results.

There are 6 ways so that all three match.

Hence, the probability p that they match is

p = 1/36 = 0.027777778

Now, note that

P(more than 10) = 1 - P(at most 10)

Now, using a table/technology for cumulative negative binomial distribution,

P(at most 10) = 0.266464767

Thus,

P(more than 10) = 1 - 0.266464767 = 0.733535233 [answer]

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Now, note that

P(more than 20|more than 10) = P(more than 20) / P(more than 10)

As

P(more than 20) = 1 - P(at most 20)

Now, using a table/technology for cumulative negative binomial distribution,

P(at most 20) = 0.446552521

Thus,

P(more than 20) = 1 - 0.446552521 = 0.553447479   

Thus,

P(more than 20|more than 10) = P(more than 20) / P(more than 10) = 0.553447479/0.733535233 = 0.754493382 [ANSWER]