Conditional Probability (1) by using statistics.pptx

A text describing probability

1. Conditional Probability The probability we assign to an event can change if we know that some other event has occurred. This idea is the key to many applications of probability.

2. The probability of an event based on the fact that some other event has occurred, will occur, or is occurring. P(B|A ) = Conditional Probability The probability of event B occurring given that event A has occurred is usually stated as “ the conditional probability of B, given A ; P(B|A ) Conditional Probability – Events Involving “And”

3. Example: Given a family with two children, find the probability that both are boys, given that at least one is a boy. P({gb, bg, bb}) = Conditional Probability P(A and B) = A = at least one boy Conditional Probability – Events Involving “And” P(A) = 3/4 P({gb, bg, bb}  {bb}) = P({bb}) = = 1/3 1/4 Conditional Probability P(B|A ) = 3/4 1/4 B = both are boys S= {gg, gb, bg, bb} A = {gb, bg, bb} B = {bb} =

4. Example: A single card is randomly selected from a standard 52-card deck. Given the defined events A and B, A: the selected card is an ace, B: the selected card is red, find the following probabilities. a) P(B) = Independent Events b) P(A and B) = a) P(B) b) P(A and B) c) P(B|A ) Conditional Probability – Events Involving “And” = 1/2 P({Ah, Ad, Ac, As}  {all red}) = P({Ah, Ad}) = 2/52 Events A and B are independent as P(B) = P(B|A ). c) P(B|A ) = P(A) P(A and B) 4/52 2/52 1/2 = = 52 26

5. Consider tossing a fair six-sided die once and define events A = {2, 4, 6}, B = {1, 2, 3}, and C = {1, 2, 3, 4}. Calculate the following: P(A) P(A|B) P(A|C) Are A and B dependent or independent? Explain Are A and C dependent or independent? Explain

6. If A and B are any two events then P(A and B) = P(A)  P(B|A ) Multiplication Rule of Probability - Events Involving “And” 11.3 – Conditional Probability – Events Involving “And” P(A and B) = P(A)  P(B) If A and B are independent events then A jar contains 4 red marbles, 3 blue marbles, and 2 yellow marbles. What is the probability that a red marble is selected and then a blue one without replacement? P(Red and Blue) = P(Red)  P( Blue|Red ) = 4/9  3/8 = 12/72 = 1/8 = 0.1667 Example:

7. Multiplication Rule of Probability - Events Involving “And” 11.3 – Conditional Probability – Events Involving “And” A jar contains 4 red marbles, 3 blue marbles, and 2 yellow marbles. What is the probability that a red marble is selected and then a blue one with replacement? P(Red and Blue) = P(Red)  P(Blue) = 4/9  3/9 = 12/81 = 4/27 = 0.148 Example:

8. Example: A number from the sample space S = {2, 3, 4, 5, 6, 7, 8, 9} is randomly selected. Given the defined events A and B, A: selected number is odd, and B: selected number is a multiple of 3 find the following probabilities. a) B = {3, 6, 9} Conditional Probability b) P(A and B) = a) P(B) b) P(A and B) c) P(B|A ) Conditional Probability – Events Involving “And” P(B) = 3/8 P({3, 5, 7, 9}  {3, 6, 9}) = P({3, 9}) = 2/8 = 1/4 c) Probability of B given A has occurred: P(B|A ) = P(A) P(A and B) 4/8 1/4 1/2 = =

9. The table shows the results of a class survey. Find P (own a pet | female) Conditional Probability The condition female limits the sample space to 14 possible outcomes. Of the 14 females, 8 own a pet. Therefore, P (own a pet | female) equals . 8 14 yes no female 8 6 male 5 7 Do you own a pet? 14 females; 13 males

10. The table shows the results of a class survey. Find P (wash the dishes | male) Conditional Probability The condition male limits the sample space to 15 possible outcomes. Of the 15 males, 7 did the dishes. Therefore, P (wash the dishes | male) 7 15 yes no female 7 6 male 7 8 Did you wash the dishes last night? 13 females; 15 males

11. Using the data in the table, find the probability that a sample of not recycled waste was plastic. P (plastic | non-recycled) The given condition limits the sample space to non-recycled waste. Material Recycled Not Recycled Paper 34.9 48.9 Metal 6.5 10.1 Glass 2.9 9.1 Plastic 1.1 20.4 Other 15.3 67.8 The probability that the non-recycled waste was plastic is about 13%. = 20.4 156.3 0.13 A favorable outcome is non-recycled plastic. P (plastic | non-recycled) = 20.4 48.9 + 10.1 + 9.1 + 20.4 + 67.8 Let’s Try One

12. Conditional Probability Researchers asked people who exercise regularly whether they jog or walk. Fifty-eight percent of the respondents were male. Twenty percent of all respondents were males who said they jog. Find the probability that a male respondent jogs. Relate: P ( male ) = 58% P ( male and jogs ) = 20% Define: Let A = male. Let B = jogs. The probability that a male respondent jogs is about 34%. Write: P ( A | B ) = P ( A and B ) P ( A ) = Substitute 0.2 for P ( A and B ) and 0.58 for P ( A ). 0.344 Simplify. 0.2 0.58

13. On Monday April 15, 1912, the RMS Titanic sank in the North Atlantic Ocean on its maiden voyage from Southampton to New York. The data concerned with this epoch event in history is presented in Table Q3b. Use it to answer the following questions .

14. Table on Titanic Mortality men women boys girls total survived 332 318 29 27 706 died 1360 104 35 18 1517 total 1692 422 64 45 2223 Assuming that 1 person is randomly selected from the 2223 people aboard the Titanic , Find P (selecting a man or a boy). Find P (selecting a man or someone who survived). Find P (woman or child | survivor) Find P (survived | man) Find P (man | survived)

15. Using Tree Diagrams Jim created the tree diagram   after examining years of weather observations in his hometown. The diagram shows the probability of whether a day will begin clear or cloudy, and then the probability of rain on days that begin clear and cloudy. a. Find the probability that a day will start out clear, and then will rain. The path containing clear and rain represents days that start out clear and then will rain. P (clear and rain) = P (rain | clear) • P (clear) = 0.04 • 0.28 = 0.011 The probability that a day will start out clear and then rain is about 1%.

16. Conditional Probability (continued) b. Find the probability that it will not rain on any given day. The paths containing clear and no rain and cloudy and no rain both represent a day when it will not rain. Find the probability for both paths and add them. P (clear and no rain) + P (cloudy and no rain) = P (clear) • P (no rain | clear) + P (cloudy) • P (no rain | cloudy) = 0.28(.96) + .72(.69) = 0.7656 The probability that it will not rain on any given day is about 77%.

17. Let’s Try One A survey of Pleasanton Teenagers was given. 60% of the responders have 1 sibling; 20% have 2 or more siblings Of the responders with 0 siblings, 90% have their own room Of the respondents with 1 sibling, 20% do not have their own room Of the respondents with 2 siblings, 50% have their own room Create a tree diagram and determine P (own room | 0 siblings) P (share room | 1 sibling)

18. 60% of the responders have 1 sibling; 20% have 2 or more siblings Of the responders with no siblings, 90% have their own room Of the respondents with 1 sibling, 20% do not have their own room Of the respondents with 2 siblings, 50% have their own room Create a tree diagram and determine P (own room | 0 siblings) P (share room | 1 sibling)

19. The voters in a large city are 40 % white, 40 % black, and 20 % Hispanic. A black mayoral candidate anticipates attracting 30 % of the white vote, 90 % of the black vote, and 50 % of the Hispanic vote. Draw a tree diagram with probabilities for the race (white, black, or Hispanic) and vote (for or against the candidate) of a randomly chosen voter. What percent of the overall vote does the candidate expect to get ? Lactose intolerance causes difficulty digesting dairy products that contain lactose (milk sugar). It is particularly common among people of African and Asian ancestry. In the United States, 82 % of the population is white, 14 % is black, and 4 % is Asian. Moreover, 15 % of whites, 70 % of blacks, and 90 % of Asians are lactose intolerant. Construct a tree diagram for the data given; What percent of the entire population is lactose intolerant? What percent of people who are lactose intolerant are Asian?

20. Question 1 In a class of a certain school, 20 of the students study French, 12 study Geography, and 9 study History. 12 students study only French, 4 study only History and 5 study only Geography. Every student studies at least one of the three subjects but no student studies all three. How many students are there in the class? If a student is selected at random from the class, find the probability that the person chosen studies b oth History and Geography e xactly two subjects c) If a student is selected at random from the list of all those who do not study French, find the probability that the person chosen studies b oth History and Geography o nly Geography

21. Question 2 Suppose that A and B are events with probabilities: P(A)=1/3, P(B)=1/4, P(A ∩ B)= 1/10. Find each of the following: 1. P(A | B) 2. P(B | A) 3 . P(A’ | B’)