Understanding Probability: Sample Spaces & Events

1. Chapter 4.1 Sample Spaces and Probability

2. Probability • “Life is a school of probability” ~ Walter Bagehot • “The only two sure things are death and taxes” ~ cynical philosopher • “Statistically, the probability of any one of us being here is so small that you'd think the mere fact of existing would keep us all in a contented dazzlement of surprise” ~ Lewis Thomas

3. Probability • Probability can be defined as the chance of an event occurring • A probability experiment is a chance process that leads to well-defined results called outcomes • An outcome is the result of a single trial of a probability experiment • A sample space is the set of all possible outcomes of a probability experiment

4. Examples of Sample space and probability experiments

5. Sample space • Find the sample space for rolling two dice

6. Sample space • Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl.

7. Tree diagram • A tree diagram is a device consisting of line segments emanating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment

8. Make a tree diagram the family with 3 children

9. Event • An event consists of a set of outcomes of a probability experiment • An event with one outcome is called a simple event • A compound event consists of two or more outcomes or simple events

10. The three types of Probability • Classical or theoretical • Empirical or Relative frequency or Experimental • Subjective

11. Classical probability • Classical probability uses sample spaces to determine the numerical probability that an event will happen. You do not actually perform the experiment to determine the probability. • Classical probability assumes that all outcomes in the sample space are equally likely to occur.

12. Formula for classical probability • The probability of any event E is the number of outcomes in E divided by the total number of outcomes in the sample space. • Denoted by

13. Range of values for probability • What is the probability that the sun will rise tomorrow? • What is the probability that Indiana Jones will come crashing through our window? • What is the probability of getting heads when a coin is flipped?

14. Rounding rules • Probabilities should be expressed as reduced fractions or rounded to two or three decimal places. • When the probability of an event is an extremely small decimal, round to the first nonzero digit after the decimal point. • For example: 0.0000587 would be rounded to 0.00006

15. Equally likely events • Equally likely events are events that have the same probability of occurring. • For example, flipping a coin and getting heads or tails

16. Find the probability of each event using a single die • P(3) • P(even) • P(odd) • P(prime) • P(4 or 5) • P(value less than 7)

17. Find the probability for each event using a standard deck of cards • P(jack) • P(heart) • P(black ten) • P(six of clubs) • P(3 or 6) • P(3 or diamond)

18. Gender of children • If a family had three children, find the probability that exactly two of the three children are girls. • Find the probability that at least one of the three children is a boy.