+ Independence
+ Compound Events When working with probabilities we often perform more than one event in a sequence - this is called a compound probability. Compound probabilities are more complex than a single event probability to compute because the first event might affect the probability of the second event happening.
+ Independence The independence of 2 events is determined by asking the following question: Does the occurrence of the 1 st event affect the probability of the 2 nd event occurring? If it doesn t - INDEPENDENT EVENTS If it does NOT INDEPENDENT EVENTS
+ Are these events independent? Flipping a coin 2 times.
+ Are these events independent? Flipping a coin 2 times. INDEPENDENT The probability of getting a head on a single flip of a coin is ½. If you flip the coin and get a head, the second flip s probability of getting a head is still ½ because the results of the first flip does not in any way affect the second flip. The second flip has the exact same probability as if it was the first flip, ½.
+ Are these events independent? Pulling 2 cookies out of a jar containing 7 chocolate chip and 3 peanut butter cookies.
+ Are these events independent? Pulling 2 cookies out of a jar containing 7 chocolate chip and 3 peanut butter cookies. NOT INDEPENDENT When pulling the 1 st cookie out, the probability of getting a chocolate chip is 7/10 and the probability of getting a peanut butter cookie is 3/10. Is the probability of getting a peanut butter cookie still 3/10, if you first pick out a chocolate chip cookie and eat it? Of course not, the probability of getting a peanut butter cookie now is 3/9 because a chocolate chip cookie is gone from the jar. The second selection is affected by the first selection, thus these two events are NOT INDEPENDENT.
+ Compound Probabilities of Independent Events What is the probability of rolling a 6 and then getting heads on a coin flip? Tree Diagram: P(A and B) =
+ Compound Probabilities of Independent Events What is the probability of rolling a 6 and then getting heads on a coin flip? Sample Space = {1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T} Event A = {6H, 6T} Event B = {1H, 2H, 3H, 4H, 5H, 6H} P(A) = P(B)= Events A and B = { } P(A and B) =
+ Compound Probabilities of Independent Events Want a faster way? When two events, A and B, are independent, the probability of both occurring in sequence is: P(A and B) = P(A) P(B)
+ Compound Probabilities of Independent Events Want a faster way? When two events, A and B, are independent, the probability of both occurring in sequence is: In other words: P(A and B) = P(A) P(B) To find the probability of 2 independent events that occur in sequence, find the probability of each event occurring separately, and then multiply the probabilities.
+ Compound Probabilities of Independent Events When two events, A and B, are independent, the probability of both occurring in sequence is: Back to the example: P(A and B) = P(A) P(B) What is the probability of rolling a 6 and then getting heads on a coin flip?
+ Examples What is the probability of rolling a 6 and then rolling a 5?
+ Examples Given a bag of marbles with 3 red, 2 green and 5 yellow. What is the probability of choosing a red, replacing it, and then choosing a green?
+ Examples What is the probability of getting heads on a coin flip and then choosing a purple marble from a bag that has 2 purple, 1 green, and 2 orange marbles?
+ Examples A true and false question is followed multiple choice question with possible four answers (1 correct & 3 wrong). What is the probability of getting both questions correct, P(CC)?
+ Mutually Exclusive vs. Independence A common misunderstanding is that independence is the same thing as being mutually exclusive. Reason for the confusion? To be independent in a typical English language context means to be alone or separate which is basically what we understand mutually exclusive to mean. This definition of independence is NOT the mathematical one. Vowels vs. Consonants in Scrabble letter bag: Independence in math is about whether one event affects another event s probability or not. Mutually exclusive sets are those that don t share any elements and independent sets are those that don t impact each other s probabilities. Mutually exclusive is about the sharing of elements, and independence is about affecting each other. Roll a die twice:
+ Testing Mutual Exclusivity & Independence To determine if two events are mutually exclusive check for shared outcomes in sample space. To determine if 2 events are independent, check using our formula: P(A and B) = P(A) P(B) Why does this work? We proved this formula only applies to INDEPENDENT EVENTS, so if your numbers result in a true equation, the events must be independent, if it results in a false equation, the events are not independent.
+ Testing Mutual Exclusivity & Independence Are events A & B mutually exclusive? YES NO Are events A & B independent? YES NO
+ Testing Mutual Exclusivity & Independence Are events A & B mutually exclusive? YES NO Are events A & B independent? YES NO
+ Finding missing probabilities given 2 independent events. P(A and B) = 0.3 P(A and Not B) = 0.2 P(A) = P(B) =
+ 1 HUGE clue in knowing whether or not two events are independent The terms replacement and no replacement get used a lot in compound probabilities problems because they describe what you did with the first thing that you selected did you put it back or did you keep it? REPLACEMENT Because the item is replaced, it resets the event back to the original arrangement and no probabilities are altered. Thus REPLACEMENT tells us that the events are INDEPENDENT. NO REPLACEMENT Because the item is NOT replaced, the probabilities are altered. Thus NO REPLACEMENT tells us that the events are NOT INDEPENDENT.
+ Your Assignment Independence Wkst