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CONDITIONAL PROBABILITY Conditional Probability • Knowledge that one event has occurred changes the likelihood that another event will occur. • Denoted: P(A|B) • The probability of A given than B has already occurred. Conditional Probability P A B P A / B P B Ex: Population has .1% of all individuals having a certain disease. A test is available and 80% of those who test positive actually have the disease. E = the individual has the disease F = the individual’s diagnostic test is positive P(E) = P(E | F) = Before the test, the occurrence of E was highly unlikely, after the test turns out positive it’s highly likely. Example: Titanic 1st Class 2nd Class 3rd Class Survived 203 118 178 499 Died 122 167 528 817 325 285 706 1316 P(3rd class) P(3rd class | survived) P(1st class | survived) P(died | 2nd class) Students at the University of New Harmony received 10,000 course grades last semester Total 6300 1600 2100 Total 3392 2952 3656 10000 • Find P(lower than B) • Find P(E|Low) and P (Low|E). • Which of the above tells you whether this college’s engineering students tend to earn lower grades than students in liberal arts. Find the probability of drawing a 3 of diamonds if you already know that it’s a red card. Find probability of a family having 2 girls given that they have at least 1 girl. A consumer organization estimates that over a 1-year period 17% of cars will need to be repaired once, 7% will need repairs twice, and 4% will require 3 or more repairs. • Probability that a car chosen at random will need a. No repairs b. No more than one repair c. Some repairs You randomly draw a card at random from a standard deck of 52 cards. • P(heart | red) • P(red | heart) • P(ace | red) • P(queen | face card) 70% of kids who visit a doctor have a fever, and 30% of kids with a fever have sore throats. What’s the probability that a kid who goes to the doctor has a fever and a sore throat? In a class of 45 students 18 like apples and 32 like bananas and 5 dislike both fruits. If a students is randomly selected, find the probability that the student: • likes both fruit • Likes at least one fruit • Likes Bananas given that they like apples • Dislikes apples given that they like bananas. Independent Events • Two events A and B are independent if the occurrence of one event has no effect of the chance that the other event will happen. • In other words, events A and B are independent if P A / B P A and PB / A PB Using the blocks: • What’s the probability of drawing a green? • If I replace the block and draw again what is the probability that the second block is green? Using the blocks: • What’s the probability of drawing a green? • If I do not replace the block and draw again what is the probability that the second block is green? So…. • Sampling with replacement is ____________ • Sampling without replacement is ________________ If I roll 2 dice, what’s the probability that I get a three on both? Independent or Dependent? Is there a relationship between gender and handedness? To find out, we used CensusAtSchool’s Random Data Selector to choose a SRS of 50 Australian high school students who completed a survey. Dominant hand Gender Right Left Total Male 20 3 23 Female 23 4 27 Total 43 7 50 If two events are not independent, does that mean than there actually is a relationship between the two variables? • No…not necessarily Independent or not independent? • Shuffle a standard deck of cards, and turn over the top card. Put it back in the deck, shuffle again, and turn over the top card. Define events • A: first card is a heart • B: second card is a heart Independent or not independent? • Shuffle a standard deck of cards, and turn over the top two cards, one at a time. Define the events • A: first card is a heart • B: second card is a heart Independent or not independent? • The 28 students in a class completed a brief survey. One of the questions asked whether each student was right- or left-handed. Choose a student from the class at random. The events of interest are “female” and “right-handed.” Gender Handedness Female Left Right 3 18 Male 1 6 Multiplication Rule • Dependent: • Independent: P(King and then King) with replacement. P(king & King) without replacement. P(king and 4) without replacement P(E)=.2 P(F)=.3 • Find P(E and F) if they are independent. If P(A) = 0.2 and P(B) = .4 and P(A B) = p. Find p if • A and B are mutually exclusive • A and B are independent Using a tree diagram, what’s the probability of getting two heads when you toss a fair coin? Suppose that 60% of all customers of a large insurance agency have automobile policies with the agency, 40% have homeowner’s policies, and 25% have both types of policies. If a customer is randomly selected, what is the probability that he or she has at least one of these two types of policies with the agency? (Look at Venn Diagram) The probability of purchasing a Sony DVD player is 0.70. The probability of an extended warranty being purchased when a Sony DVD player is bought is 0.20. Find the probability that a person buys a Sony DVD and the extended warranty. Homework • Page 329 (63, 65, 67, 69, 71, 73, 75, 79, 106)