Thread: Cardinality of events?

I do not understand how to do these problems.

1. 1 dice is rolled five times, find the probability that the sequence is increasing (ex. 12345, 12346, 23456, etc.)
2. 1 dice is rolled five times, find the probability that the sequence contains exactly two numbers (11234, 22314, etc.)

I can get the cardinality of the sample space. (6^5)


I'm mostly having a hard time with the cardinality of the events. There are so many possibilities for each of these, it's impossible to write them out on paper. How do I get the cardinality of all these events?



This assignment is due soon and for the most part I understand it. But I do not understand how to get these problems with the large event cardinalities.

Hello, jimpo2!

1. A die is rolled five times.
Find the probability that the sequence is increasing (ex. 12345, 12346, 23456, etc.)

This one is simple if you think about it . . .

Of the possible outcomes,
. . only five are in increasing order: .

Therefore: .

2. A die is rolled five times.
Find the probability that the sequence contains exactly one pair: (11234, 22314, etc.)

The Pair can be any of the 6 values.

The Pair can show up in any two of the five rolls: . ways.

The probability that two rolls have that Pair is: .

The other three rolls must not match the Pair or each other: .


Therefore: .