Chapter 4 Probability

1. Chapter 4Probability The description of sample data is only a preliminary part of a statistical analysis. A major goal is to make generalizations or inferences about the target population on the basis of the information obtained from the sample data. The subject of probability is essential for understanding the reasoning that leads to such generalizations.

2. Terminology

3. Example: Toss a coin twice and record the outcome H or T for each toss. Tree diagram describes the experiment: Sample space: S = {HH, HT, TH, TT}. Let A denote the event of getting exactly one head. Then A = {HT, TH}.

4. Probability of an event

5. Methods of assigning probability • Equally likely elementary outcomes – the uniform probability model • Probability as the long-run relative frequency

6. The uniform probability model Coin example (cont.): Sample space: S = {HH, HT, TH, TT}. Let A denote the event of getting exactly one head. A = {HT, TH}. P(A) = 2/4 = .5

7. Example (genetics): Inherited characteristics are transmitted from one generation to another by genes. Genes occur in pairs and the offspring obtain their pair by taking one gene from each parent. Experiment: Cross fertilize red flower with a white flower. Result is a hybrid (pink flower). Crossing these hybrids leads to one of 4 possible gene pairs: Mendel’s law: P(pink) = ½ P(white) = P(red) = 1/4 An experiment resulted in 141, 291, 132 white, pink, and red flowers. These numbers are nearly in ratio 1:2:1 .

8. Example: Two dice are tossed. The possible outcomes are shown below: Compute the probabilities of the following events: A = {sum = 6}, B = {sum=7}, C = {sum is even}

9. Probability as the long-run relative frequency In many situations, the elementary outcomes are not equally likely. Example: one corner of a die is cut off. In those situations, repeat the experiment many times and observe the proportion of times the event occurs: Relative frequency of event A in N trials = (No. of times A occurs in N trials) /N

10. Long-run stability of relative frequency In the long-run, the relative frequency stabilizes. Example:

11. Example: