15‐5 Conditional Probability with Frequency Tables Vocabulary: Conditional Probability – the probability that an event will occur, given that another even has already occurred. Two‐way frequency table – a table that displays the frequencies of data in two different categories. Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data. The probability that an event B will occur, given that another even A has already occurred, is called a conditional probability and is written P(B|A). You read P(B|A) as “the probability of event B, given event A.” Example: You randomly select two marbles, one at a time, from a bag containing 3 red marbles and 5 green marbles. If your first marble is red, then 2 of the remaining 7 marbles are red, so P(2nd marble is red|1st marble is red) = Using a two‐way frequency table The following table shows data about student involvement in extracurricular activities at a local high school. What is the probability that a randomly chosen student is a female who is not involved in extracurricular activities? Involved in Activities Male 112 Female 139 Totals 251 To find the probability, calculate the relative frequency. Relative frequency = Not involved in Activities 145 120 265 Totals 257 259 516 Finding Probability Respondents of a poll were asked whether they were for, against, or had no opinion about a bill before the state legislature that would increase the minimum wage. What is the probability that a randomly selected person is over 60 years old, given that the person had no opinion on the state bill? Age Group For Against No Opinion Totals 18‐29 310 50 20 380 30‐45 200 30 10 240 46‐60 120 20 30 170 Over 60 150 20 40 210 Totals 780 12 100 1000 The condition that the person selected has no opinion on the minimum‐wage bill limits the total outcomes to the 100 people who had no opinion. Of those 100 people, 40 respondents were over 60 years old. P(over 60|no opinion) = Using Relative Frequencies A Company has 150 sales representatives. Two month after a sales seminar, the company vice‐
president made the table of relative frequencies based on sales results. What is the probability that someone who attended the seminar had an increase in sales? Increased sales No Increase in Sales Totals Method 1 Attended Seminar 0.48 0.32 0.8 Did not attend Seminar 0.02 0.18 0.2 Method 2 Find frequencies first. Use relative frequencies. Find the number of people who attended the seminar and had increased sales: P(increased sales|sales seminar)= Find the number of people who attended the seminar: Find P(increased sales|sales seminar): Totals 0.5 0.5 1 15‐6 Conditional Probability Formulas For any two events A and B, the probability of B occurring, given that event A has occurred, is |
If A and B are dependent events, then ,
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0 |
Using Conditional Probability In a study designed to test the effectiveness of a new drug, half of the volunteers received the drug. The other half of the volunteers received a placebo, a tablet or pill containing no medication. The probability of a volunteer receiving the drug and getting well was 45%. What is the probability of someone getting well, given that he receives the drug? Identify the probabilities: P(B|A) = P(getting well, given taking the new drug) = P(A) = P(taking the new drug) = P(A and B) = P(taking the new drug and getting well) = Find P(B|A): |
Comparing Conditional Probabilities In a survey of pet owners, 45% own a dog, 27% own a cat, and 12% own both a dog and a cat. What is the conditional probability that a dog owner also owns a cat? What is the conditional probability that a cat owner also owns a dog? P(cat|dog) = P(dog|cat) = Selecting with replacement You choose a colored marble randomly from a bag that contains 4 blue marbles, 3 red marbles, 5 purple marbles, 2 green marbles, and 1 white marble. You replace the first marble, and then choose again. What is the probability that you choose a blue marble and then a red marble? P(blue) = P(red) = P(blue and red) = Selecting without replacement You choose a colored marble randomly from a bag that contains 4 blue marbles, 3 red marbles, 5 purple marbles, 2 green marbles, and 1 white marble. Without replacing the first marble, you select a second marble. What is the probability of selecting a blue marble and then a red marble? P(blue) = P(red after blue) = P(blue then red) = Using a Tree Diagram A college reported that 70% of its freshmen had attended public schools, 60% of the freshmen who had attended public schools graduated within 5 years, and 80% of other freshmen graded within 5 years. What percent of freshmen graduated within 5 years? You can use a tree diagram to organize the information: Graduate Public
Not Graduate
P(Public and Graduate) = Graduate Other
P(Other and Graduate) = P(Graduate) = Not Graduate