Class 13 Overview: Probability part 2. Two Way Tables

Transcription

1 Class 13 Overview: Probability part 2 Two Way Tables Example 1: Using data from , the Steelers won 56 / 80 or 70% of their games. Of the games won, 31 / 56 or 55% were played at home. Of the games lost, 9 / 24 or 38% of them were played at home. Example 2: A class of 100 college students was surveyed about whether or not they had decided on a major. Of those surveyed 45 have decided on a major. Of those who have decided, 27 are female. There were 60 females in the survey. Example 3: At a medical center there are 18 doctors, 12 of whom are male and there are 10 nurses, 4 of whom are male.

2 Probability Trees Example 1: The Los Angels Angels play half of their games at home and half away. Of their home games they won 55% and lost 45%. They won 46% of their away games and lost 54%. Example 2: 30% of Math 219 students use graphing calculators. 80% of those with graphing calculators passed exam 1. 40% of those with no graphing calculator did not pass exam 1.

3 Principles Value for the Probability of an Event: The probability of an event, which informs us of the likelihood of it occurring, can range anywhere from 0 (indicating that the event will never occur) to 1 (indicating that the event is certain). Sum of the Probabilities of all Possible Outcomes: The sum of all of the probabilities for all of the possible outcomes for an experiment is 1. Example: As was previously discussed, all human blood can be typed as O, A, B, or AB. In addition, the frequency of these blood types varies by ethnic and racial groups. A person in the U.S. is chosen at random. What is the probability of the person having blood type A? Complement of an Event (definition) The complement of an event is everything else that can happen other than the event in question. Example: Name the complement of each event. a.) drink at least 8 glasses of water daily b.) study more than 10 hours per week c.) study exactly 12 hours per week

4 Example: On the "Information for the Patient" label of a certain anti-depressant it is claimed that based on some clinical trials, there is a 14% chance of experiencing sleeping problems known as insomnia (denote this event by I) there is a 26% chance of experiencing headaches (denote this event by H), and there is a 35% chance of experiencing at least one of these two side effects (denote this event by L) a.) What is the probability that a patient taking this drug will not experience insomnia? b.) In this context, the complement of event L, "not L" is the event that: c.) The probability of "not L" is: And vs. Or We will start by creating a list of qualities that an ideal presidential candidate should have If we use the word AND, we might say we want a candidate to be: Use the same list and use the word OR, what does the sentence now mean:

5 Disjoint or Mutually Exclusive Events (definition) Two events are called disjoint or mutually exclusive if they cannot occur at the same time. In symbols, we could say that if event A and event B are disjoint, then P(A and B) = zero. Recall that if something is impossible its probability is zero. Consider these scenarios: 1. Is it possible for a statistics professor to grade fairly and also be a hard grader? 2. Is it possible for a statistics professor to know the material and to make mistakes while doing problems? 3. Is it possible for a statistics professor to state his/her expectations clearly and also to be vague about his/her expectations? Example: Decide if the events are mutually exclusive. i.) A = it is rainy; B = it is windy iv.) A=wear sandals; B=wear running shoes ii.) A=accepted to USC; B=accepted to UCLA v.) A=divorced; B=never have been married iii.) A=male; B=pierced ears vi.) A= vehicle is a car; B=vehicle is red

6 Recall the example with the Pittsburgh Steelers generated the following tree. We have added the outcomes. We now wish to be able to calculate the following probabilities: 1. P(home) 2. P(away) 3. P(win or home) 4. P(lose or home) 5. P(win or away) 6. P(lose or away)

7 Example: We are given that 60% of those surveyed were children and the rest were adults. Of the children, 72% liked the movie, but only 34% of the adults liked it. a.) Draw the tree or two way that represents this situation: b.) P(like the movie) c.) P(child or dislike the movie) Example: The data which generated this two-way table asked a random sample of 1200 U.S. college students to describe their body-image as underweight, overweight, or about right. a.) P(Female) b.) P(Overweight) c.) P(Overweight) d.) P(Female or About Right) e.) P(About Right or Underweight)

8 Example: It is vital that a certain document reach its destination within one day. To maximize the chances of ontime delivery, two copies of the document are sent using two services, service A and service B. It is known that the probabilities of on-time delivery are: 0.90 for service A, thus, P(A) = for service B, thus, P(B) = for both services being on time, thus, P(A and B) = 0.75 We wish to find the probability of on-time delivery of the document. Finding the Probability of at least one of A patient with blood type O desperately needs a blood transfusion. Since a person with blood type O can receive blood only from another person who has blood type O, the blood bank decides to choose 10 donors at random and hope that at least one of them has blood type O. Find P(at least one of the 10 donors has blood type O).

9 Example: A machine makes computer chips. The probability that a chip is defective is.01. If 20 chips are made, what is the probability that at least one is defective? Example: