We should appreciate cherish and cultivate blessings 1
We should appreciate, cherish and cultivate blessings. 1
Chapters 10, 12: Probability l. Sample space/event l. Probability models l. Basic probability rules l. Random variables 2
Random Phenomenon & Probability 3 l A random phenomenon is one in which the outcome is unpredictable. The outcome is unknown until we observe it l The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions
Example 1: Traffic Jam on the Golden Bridge l l 4 < = 20 mph traffic jam How often will a driver encounter traffic jam on the golden bridge during 7 -9 am weekdays? __ out of 100 times.
Example 2: Coin Flipping l 5 What is the probability that a flipped coin shows heads up?
Two Ways to Determine Probability 6 l Making an assumption about the physical world and use it to find proportions l Observe outcomes and find its proportions in a very long series of repetitions
Probability Models 7 l Sample space: the collection of all possible outcomes of a random phenomenon l An event is an outcome or a set of outcomes of a random phenomenon; i. e. a subset of the sample space l A probability model consists of two parts: a sample space S and a way of assigning probabilities to events; i. e. a model for distribution ** what are the sample space/event of the examples?
Rules for a Probability Model 8 l The probability P(A) of any event A is between 0 and 1 l The probability P(S) of sample space is 1
Building a Probability Model 9 l (Theoretical way) Making an assumption about the physical world and use it to build the model l (Data way) Measuring a representative sample and observing proportion of the sample that fall into various outcomes l Do example 2 and then example 3
Example 3: Employment l What is the probability that a CSUEB graduate gets hired within 3 months after graduation? Between 3 to 6 months? More than 6 months? ** Data: In a representative sample of 200 graduates, 150, 30, 20 of them got hired within 3, 3 -6, and > 6 months, respectively. 10
Probability of an Event l The sum of the probabilities of outcomes in the event ** Revisit the examples 2, 3 and find the events & their probabilities 11
Mutually Exclusive Events l 12 Two events are mutually exclusive if they do not contain any of the same outcomes. They are also called disjoint.
Basic Probability Rules (p. 269) l Rule 3: If two events A and B are disjoint, then P(A or B) = P(A)+P(B) l Rule 4: For any event A, P(A does not occur) = 1 - P(A) ** The event (A or B) occurs if either A or B or both occur. 13
Probability Models 14 l A probability model with a finite sample space is called discrete Example: problem 10. 11 (p. 272) l A continuous probability model assigns probabilities as areas under a density curve. The area under the curve and above any range of values is the probability of an outcome in that range Example: randomly pick a number between 0 and 1
Random Variables l A random variable is a variable whose value is a numerical outcome of a random phenomenon l The distribution of a random variable X tells us what values X can take and how to assign probabilities to those values Example: 15 1. # of dots (problem 10. 11) and 2. height of a young lady (example 10. 9)
Independent Events l 16 Two events are independent if the probability that one event occurs stays the same, no matter whether or not the other event occurs.
Conditional Probability The conditional probability of the event B given the event A, denoted P(B|A), is the long -run relative frequency with which event B occurs when circumstances are such that event A also occurs. ** event A = age 21+; event B = female ** Ask students for age and find P(B|A) l 17
Basic Rules for Finding Probabilities l l 18 P(A or B) = P(A) + P(B) - P(A and B) k disjoint events: P(A 1 or A 2 or … or Ak) = P(A 1) + P(A 2) + … + P(Ak) P(A and B) = P(A)P(B|A) k independent events: P(A 1 and A 2 and … and Ak) = P(A 1)P(A 2)…P(Ak)
Steps for Finding Probabilities 1. 2. 3. 4. 5. 19 Identify random phenomenon Identify the sample space Build the probability model as much as you can Specify the event for which the probability is wanted Use the probability model from step 3 and the probability rules to find the probability of interest
Tree diagrams l 20 For a sequence of events, when conditional probabilities for events based on previous events are known
Example: l 21 People are classified into 8 types. For instance, Type 1 is “Rationalist” and applies to 15% of men and 8% of women. Type 2 is “Teacher” and applies to 12% of men and 14% of women. Each person fits one and only one type.
• What is the probability that a randomly selected male is “Rationalist”? “Teacher”? Both? • What is the probability that a randomly selected female is not a “Teacher”? 22
Suppose college roommates have a particularly hard time getting along with each other if they are both “Rationalists. ” A college randomly assigns roommates of the same sex. What proportion of male roommate pairs will have this problem? What proportion of female roommate pairs will have this problem? Assuming that half of college roommate pairs are male and half are female. What proportion of all roommate pairs will have this problem? 23