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What three things do statistical experiments have in common?
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What is a good example of a statistical experiment?
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Stats - 4.1-4.4
Concepts of Probability
| Question | Answer |
|---|---|
| What three things do statistical experiments have in common? | -The experiment can have more than one possible outcome. -Each possible outcome can be specified in advance. -The outcome of the experiment depends on chance. |
| What is a good example of a statistical experiment? | A coin toss has all the attributes of a statistical experiment. There is more than one possible outcome. We can specify each possible outcome (i.e., heads or tails) in advance. And there is an element of chance, since the outcome is uncertain. |
| What is a sample space? | A sample space is a set of elements that represents all possible outcomes of a statistical experiment. |
| What is a sample point? | A sample point is an element of a sample space. |
| What is an event? * | An event is a subset of a sample space - one or more sample points. |
| What is an example of an event? | |
| What are the different types of events? | 1. Mutually exclusive 2. Independent |
| When are events mutually exclusive? * | Two events are mutually exclusive if they have no sample points in common. |
| When are events independent? * | Two events are independent when the occurrence of one does not affect the probability of the occurrence of the other. Ex. If i roll a dice twice, the first roll does not influence the second roll. |
| What is the difference between independent and mutually exclusive? * | |
| What is a null/empty set? | A set that contains no elements is called a null set or an empty set. |
| What is a set? | A set is a well-defined collection of objects. |
| What is an element? | Each object in a set is called an element of the set. |
| When are two sets equal? | Two sets are equal if they have exactly the same elements in them. The order of the elements does not matter. |
| What is a subset? * | If every element in Set A is also in Set B, then Set A is a subset of Set B. |
| How are sets represented? | A set is usually denoted by a capital letter, such as A, B, or C. |
| How are elements represented? | An element of a set is usually denoted by a small letter, such as x, y, or z. |
| How is a null set represented? | The null set is denoted by {∅} |
| What is a union?* | The union of two sets is the set of elements that belong to one or both of the two sets. Thus, set Z is the union of sets X and Y. If W = {2}; X = {1, 2}; Y= {2, 3, 4}; and Z = {1, 2, 3, 4}. |
| How is a union represented? | U |
| What is an intersection?* | The intersection of two sets is the set of elements that are common to both sets. Thus, set W is the intersection of sets X and Y. If W = {2}; X = {1, 2}; Y= {2, 3, 4}; and Z = {1, 2, 3, 4}. |
| How is intersection represented? | Symbolically, the intersection of X and Y is denoted by X ∩ Y. An arch. An upside-down u. |
| What can we do to make counting points in a sample space easier/faster? | Fortunately, there are ways to make the counting task easier. There are three rules of counting that can save both time and effort - event multiples, permutations, and combinations. |
| What is an event multiple? * | An event multiple occurs when two or more independent events are grouped together. The first rule of counting helps us determine how many ways an event multiple can occur. |
| What is the range of the probability of a sample point? What is the sum of the probabilities of all sample points in a sample space? | -The probability of any sample point can range from 0 to 1. -The sum of probabilities of all sample points in a sample space is equal to 1. |
| What is the probability of an event? | The probability of an event is a measure of the likelihood that the event will occur. -Can range from 0 to 1. -The probability of event A is the sum of the probabilities of all the sample points in event A. |
| What is the law of large numbers? * | The idea that the relative frequency of an event will converge on the probability of the event, as the number of trials increases, is called the law of large numbers. Frequencey Event/# of trials. |
| When are two events mutually exclusive or disjoint? | Two events are mutually exclusive or disjoint if they cannot occur at the same time. |
| What is conditional probability? | The probability that Event A occurs, given that Event B has occurred, is called a conditional probability. The conditional probability of Event A, given Event B, is denoted by the symbol P(A|B). |
| What is the complement of an event? | The complement of an event is the event not occurring. The probability that Event A will not occur is denoted by P(A'). |
| What is the probability that one event AND another occur? | The probability that Events A and B both occur is the probability of the intersection of A and B. The probability of the intersection of Events A and B is denoted by P(A ∩ B). If Events A and B are mutually exclusive, P(A ∩ B) = 0. |
| What is the probability that one event OR the other occurs? | The probability that Events A or B occur is the probability of the union of A and B. The probability of the union of Events A and B is denoted by P(A ∪ B) . |
| When are two events dependent? | If the occurence of Event A changes the probability of Event B, then Events A and B are dependent. |
| When are two events independent? | On the other hand, if the occurence of Event A does not change the probability of Event B, then Events A and B are independent. |
| What is the rule of subtraction? | The probability that event A will occur is equal to 1 minus the probability that event A will not occur. P(A) = 1 - P(A') |
| What is the rule of multiplication? | The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred. P(A ∩ B) = P(A) P(B|A) |
| What is the rule of addition? | The probability that Event A or Event B occurs is equal to the probability that Event A occurs plus the probability that Event B occurs minus the probability that both Events A and B occur. P(A ∪ B) = P(A) + P(B) - P(A ∩ B)) |
| How can the rule of addition also be expressed? | Note: Invoking the fact that P(A ∩ B) = P( A )P( B | A ), the Addition Rule can also be expressed as P(A ∪ B) = P(A) + P(B) - P(A)P( B | A ) |
Created by:
Colin Reeder