Math 311

Some Examples

Example 1

• An urn contain two balls, each can be red or blue
• You randomly pick one of the balls and replace it with another one:
  • of the same color, with probability 80%
  • of the opposite color, with probability 20%

• Initially, both balls are red.

• What is the probability that the fifth selected ball is red?

Example 2

• Balls are successively distributed among 8 urns.
• Each ball is equally likely to be placed in any of the urns.

What is the probability that there will be exactly 3 non-empty urns after 9 balls are distributed?

Example 3

In a sequence of independent coin flips, let \(N\) denote the number of flips until there are 3 consecutive heads.

• Find \(\operatorname{P}(N \le 8)\)
• Find \(\operatorname{P}(N = 8)\)

Example 4

Given a Markov Chain with states 1, 2, 3, 4, and 5, with some transition probabilities

• \(P_{11} = .3\)
• \(P_{12} = .3\)
• \(P_{21} = .1\)
• \(P_{22} = .2\)

Starting at state 1, what is the probability that the chain is at state 2 after 4 steps, but never entered any of the states 3, 4, or 5?