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Prob and Stats, Oct 16 Probability – Using a Two-Way Table to Compute Probability Three Ways Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I use a table to compute simple probability, compound event probability and conditional probability? Standards: PS.SPCR.1, .1a, .1b, .1c, .6, .7, PS.SPMJ.2 Probability • Probability is, was, and always will be a number between 0 and 1 (inclusive). The Two-Way Table • A two-way table is a display of connected data, which is displayed in rows and columns • The connections of rows and columns define data relationships There are at least two connected relationships woven into every 2-way table Example The following Venn diagram and 2-way table show the relationship of a group of high school students who either play a sport, study a foreign language, or do both or neither. The First Thing (if required) • The very first thing to always check for in a 2-way table is that it is totaled (rows and column added) and that you have a grand total for the table. • If it is not done, make that your FIRST STEP in this process. Using the Two-Way Table The simple probability computation P(event) = Pertinent row or column total Table grand total • The probability of an event is the event row or column total / table grand total Examples On the first day of season practice, a high school’s football players’ hair and eye color were noted. The results are summarized below. Compute each probability: Simple Probability - Experimental On the first day of season practice, a high school’s football players’ hair and eye color were noted. The results are summarized below. Compute each probability: Compute the following probabilities for a single, randomly chosen player: P(green eyes) P(brown hair) Mutually Exclusive Events • Events A and B are mutually exclusive if they cannot occur at the same time. • Events A and B are inclusive if there is some (common) outcome that can be both events. Mutually Exclusive Events • Events A and B are mutually exclusive if they cannot occur at the same time. Sample Space A B Not occurring simultaneously means there is no overlap between the events Not Mutually Exclusive • The opposite of mutually exclusive is ‘inclusive’ – which means events can occur at the same time. • We can say inclusive or not mutually exclusive to describe this phenomenon. Inclusive Events • Events A and B are not mutually exclusive if they can occur at the same time. Sample Space A∩B A B The overlap (green section) is A ∩ B, also known as A and B Compound “OR” Probability • If two events, A and B are mutually exclusive, then P(A or B) = P(A) + P(B) • If two events, A and B are not mutually exclusive, then P(A or B) = P(A) + (B) – P(A and B) Where (A and B) are outcomes of both A and B at the same time Reminder • To use a two-way table, you must have row and column totals, and a table grand-total • The very first thing to always check for in a 2-way table is that it is totaled (rows and column added) and that you have a grand total for the table. • If it is not done, do this FIRST. • If has been done, do not do it again. The Two-Way Table • A two-way table is a display of connected data, which is displayed in rows and columns • The connections of rows and columns define data relationships Row to row or column to column relationships are mutually exclusive Row to column relationships are inclusive Row and column intersections display the value of set intersections Table Example There following table summarizes the healthcare choice of the 386 employees at the Titan Corporation and whether or not they have dependent children in their care. Dep Child No Child HMO 145 39 BC/BS 85 42 None 23 52 Table Example Compute the following probabilities: P(HMO or have dep children) P(HMO or have no coverage) Dep Child No Child Total HMO 145 39 184 BC/BS 85 42 127 None 23 52 75 Total 253 133 386 Using a Two-Way Table To Compute Compound Event Probability • Computing P(A or B) 0. If not a given, sum every row and column (including your new totals row and column) to compute every event total and a ‘grand total’ 1. Find event A on row or column heading, scribe a line 2. Find event B on row or column heading, scribe a line 3. If lines are parallel – add row or column totals, put answer over grand total, simplify (ME probability) 4. If lines cross, add row and column totals, subtract ‘cross value’ put answer over grand total, simplify (I probability) Read the Problem Decide • Upon reading any problem that contains the word or, you must Decide if the two events are mutually exclusive or inclusive If inclusive, figure the points of overlap and find that probability Apply the correct addition rule form that applies to the problem • The table lines will guide you in these decisions Examples On the first day of season practice, a high school’s football players’ hair and eye color were noted. The results are summarized below. Compute each probability: P(red hair or green eyes) P(black or brown hair) P(blond hair or hazel eyes) Conditional Probability and the Two-Way Table • If we fix a row or column with a condition, we can use it to limit the computation to that condition. When using this tool, always total the table before doing anything else. Computing Within a Condition Within the condition row or column P(event) = Pertinent entry Row or Column total • The probability of a condition occurs within a fixed row or column, and is the event value (within the condition) / row or column total Examples The following table shows the marital status of adults over 18 years of age at the 2010 census. All numbers are x 1 million: Never married Married Widowed Divorced Males 28.6 62.1 2.7 9.0 Females 23.3 62.1 11.3 12.7 Compute each probability: One adult is selected randomly P(widowed) P(male) Given that the person selected is a male, what is P(married) Given that the person selected is divorced, what is P(female) Examples The following table shows the marital status of adults over 18 years of age at the 2010 census. All numbers are x 1 million: Never married Married Widowed Divorced Males 28.6 62.1 2.7 9.0 102.4 Females 23.3 62.1 11.3 12.7 109.4 51.9 124.2 14.0 21.7 211.8 Compute each probability: One adult is selected randomly P(widowed) P(male) Given that the person selected is a male, what is P(married) Given that the person selected is divorced, what is P(female) Examples On the first day of season practice, high school football players’ hair and eye color were noted. The results are summarized below. One player is randomly chosen, compute each probability: Given that the player has blond hair, P(blue eyes) Given that the player has hazel eyes, P(red hair) Given that the player has brown eyes, P(brown hair) Three Ways of Using a Table • How do you know what is being asked – Read the problem If a simple probability statement – P(event) find the event total and divide that entry by grand total If compound event probability, P(a or b), draw lines, determine case add and subtract (if applicable), divide by grand total If conditional probability, you can tell by the word given – Given a, find P(b) – work within row or column of a, find b, divide by row or column total • No matter what, total the table if not already done so Example A student is randomly chosen: Compute: P(does not play a sport) P(does not play a sport or does not take a language) Given that the student plays a sport, P(takes a language) Class work: CW 10/16/15, 1-5 Homework: Due 10/19/15, 1-2