Stochastic Processes and Trees Chapter 3-3 Stochastic Processes and Trees

Stocahstic Processes Many experiments can be carried out in steps or stages, or they can be represented as being carried out in steps or stages. For these experiments, it is very useful to represent these steps and outcomes in tree diagrams with probabilities on the branches. We think of the main experiment as a sequence of subexperiments, one for each stage or step of the main experiment An experiment which consists of a sequence of subexperiments is called a stochastic process.

Example 1 Consider an experiment which consist of two steps. First a box is selected at random from a set of 2 boxes labeled a and b. Then an urn is selected at random from the chosen box. Box a contains 3 urns labeled A, B, and C. Box b contains 2 urns labeled D and E. Draw a tree diagram to represent the outcomes of this experiment, and assign conditional probabilities to the tree. Multiplying together the conditional probabilities on the branches path corresponding to the outcome defines the probability measure.

Example 2 Consider a three-stage experiment which uses the boxes, urns, and colored balls. Select a box, and urn and a colored ball. Suppose all selections are random. Form a tree diagram and compute the probabilities of all outcomes. There are 2 boxes (a and b). In box a, there are three urns (A, B, and C); Urn A has 2 red balls and 1 white ball; Urn B has 3 red balls and 1 white ball; Urn C has 1 red ball and 2 white balls. In box b, there are 2 urns (D and E); Urn D has 1 red ball, 2 blue balls, and 1 white ball; Urn E has 3 red balls, 1 blue ball, and 1 white ball.

Example 2 continued Find the probabilities of each event: A red ball is drawn A white ball is drawn A blue ball is drawn

Example 3 When Harry met Sally, they decided to see a movie. They could go to either of two movie complexes, one uptown showing films A, B, and X, and another downtown showing films A and B. They decided to flip a coin to pick the complex; a head is uptown (U) and a tail is downtown (D). After picking a complex, they roll a die to select a movie. If they are uptown and a 1, 2, or 3 comes up, they see movie A; if 4 or 5 comes up they see movie B; if 6 comes up they see movie X. If they are downtown, they see movie A with a 1, 2, 3, or 4 and movie B with 5 or 6. After the movie, they will have a late night snack. If they see movie A, they go to get ice cream, and if they see movie B or X, they are equally likely to go get ice cream or get sushi. 1. Find the probability they see movie B 2. Find the probability they eat sushi 3. Find the probability they see movie B and eat sushi

Example 4 A high jumper practices with a bar at a specific height. She attempts until she has 2 consecutive success, 2 consecutive misses, or a total of 4 attempts. The coach records her results. Suppose the probability she clears the bar on any attempt is 0.6. What is the probability she clears the bar exactly twice.