MATH 1680 - Statistics - Chapter 5 Probability - Section 5.4

rule of thumb for determining indepdence of events

, if the sample size, n, is less than 5% of the population size, N, we treat the events as independent. That is, if n < 0.05N, treat the events as independent.

conditional probability rule

P(F|E) = P(F and E) / P(E) THE PRObability of event F occuring, given the occurence of event E, is found by dividing the probability of E and F by the probability B, or bydividing the nmber of outcomes in E and F by the number of outcomes in E

The notation P(F E) means the probability of event ___ given event __ .

The notation P(F E) means the probability of event F given event E .

Independence using Conditional Probabilities

Two events, E&F, are independent if P(E|F) = P(E) or if P(E|F) = P(F)

In a study to determine whether preferences for self are more or less prevalent than preferences for others, researchers first asked individuals to identify the person who is most valuable and likeable to you, or favorite other. Of the 1519 individuals surveyed, 42 had chosen themselves as their favorite other. A) Suppose we randomly select 1 of the 1519 individuals surveyed. What is the probability that he or she chose themselves as their favorite other? B) If two individuals from this group are randomly selected, what is the probability that both chose themselves as their favorite other? C) Compute the probability of randomly selecting two individuals from this group who selected themselves as their favorite other assuming independence.

a) divide 42 by 1519 b) since these events are depedent bc one you choose one person then the total amount of ppl changes for the second person's probability then you multiply (42/1519) by (41/1518). c) since they are indepdendent then multiply (42/1519) by (42/1519)

how do you know the indepdence of events?

assume this if small random samples are taken from large populations

The probability that a driver who is speeding gets pulled over is 0.8. The probability that a driver gets a ticket, given that he or she is pulled over, is 0.9. What is the probability that a randomly selected driver who is speeding gets pulled over and gets a ticket?

bc of the general multiplication rule you multiply the probabilities together

The probability that a randomly selected individual in a country earns more than​ $75,000 per year is 7.5​%. The probability that a randomly selected individual in the country earns more than​ $75,000 per​ year, given that the individual has earned a​ bachelor's degree, is 21.5​%. Are the events​ "earn more than​ $75,000 per​ year" and​ "earned a​ bachelor's degree"​ independent?

no

The data in Table 5 represent the marital status and gender of U.S. residents aged 15 years and older in 2016. Table 5 M F Totals Never Married 44.1 39.0 83.1 Married 66.7 67.5 134.2 Widowed 3.5 11.4 14.9 Divorced 10.7 14.8 25.5 Totals 125. 132.7 257.7 A) Compute the probability that a randomly selected individual never married, given that the individual is male. B) Compute the probability that a randomly selected individual is male, given that the individual never married.

so compute condtional probabilities remember the dominator is the total probablility of the "given that the" event a) divide 44.1 by 125; number of never married males by total of males bc the given that the event is male b) divide 44.1 by 83.1; number of never married males by total of never married individuals bc the given that the event is individuals that have never married

Suppose that 12.2% of all births are preterm. (Preterm means that the gestation period of the pregnancy is less than 37 weeks.) Also, 0.2% of all births result in a preterm baby who weighs 8 pounds, 13 ounces or more. What is the probability that a randomly selected baby weighs 8 pounds, 13 ounces or more, given that the baby is preterm? Is this unusual? Data based on the Vital Statistics Reports.

so the probability of the weight given the probability of the baby being pretern; so divide .002 by .122 which is abt .0164 this would be an unsual event bc the P(E) is less thant .05

General Multicplication Rule

the probability that two events E and F occur is P(E and F) = P(E and F) = P(E) times P (F|E)

1) What does the notation P (F|E) represent?

this notation signifies the probability of event F, given event E -aka conditional probability, event F occurs if event E has occurded

Suppose a single die is rolled. What is the probability that the die comes up three? Now suppose that the die is rolled a second time, but we are told the outcome will be an odd number. What is the probability that the die comes up three?

use the classical method bc all outcomes are equally likely since the die is fair a) sample space, S = {1,2,3,4,5,6} thus P(3) = 1/6 b) S = {1,3,5}; P(3|roll is odd) = 1/3

Suppose that of 100 circuits sent to a manufacturing plant, 5 are defective. The plant manager receiving the circuits randomly selects two and tests them. If both circuits work, she will accept the shipment. Otherwise, the shipment is rejected. What is the probability that the plant manager discovers at least one defective circuit and rejects the shipment?

use the compliment rule *look at guided journal*