1. ## 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.

2. 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 $6^5 = 7776$ possible outcomes,
. . only five are in increasing order: . $\bigg\{12345,\:12356,\:12456,\:134456,\:23456\bigg \}$

Therefore: . $P(\text{increasing order}) \;=\;\frac{5}{7776}$

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: . ${5\choose2} = {\color{blue}10}$ ways.

The probability that two rolls have that Pair is: . $\left(\frac{1}{6}\right)^2 \:=\:{\color{blue}\frac{1}{36}}$

The other three rolls must not match the Pair or each other: . $\frac{5}{6}\cdot\frac{4}{6}\cdot\frac{3}{6} \;=\;{\color{blue}\frac{5}{18}}$

Therefore: . $P(\text{one Pair}) \;=\;6\cdot 10\cdot\frac{1}{36}\cdot\frac{5}{18} \;=\;\frac{25}{54}$