## CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes

CBSE Class 12 Commerce Mathematics Unit VI Probability : CBSE conducts the final examinations for Class 10 and Class 12 every year in the month of March. The results are announced by the end of May. The board earlier conducted the AIEEE Examination for admission to undergraduate courses in engineering and architecture in colleges across India. However the AIEEE exam was merged with the IIT-Joint Entrance Exam (JEE) in 2013. The common examination is now called JEE(Main).

CBSE also conducts AIPMT (All India Pre Medical Test) for admission to major medical colleges in India. In 2014, the conduct of the National Eligibility Test for grant of junior research fellowship and eligibility for assistant professor in institutions of higher learning was outsourced to CBSE. Apart from these tests, CBSE also conducts the central teachers eligibility test and the Class X optional proficiency test. With the addition of NET in 2014, the CBSE has become the largest exam conducting body in the world.

### CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes

CBSE Class 12 Commerce Mathematics Unit VI Probability : Cakart team members provides here CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes and other CBSE Class 12 Commerce Mathematics Complete Notes in pdf format. We provides you direct link for downloading CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes in pdf format. Download CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes and read well.

### CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes

CBSE Class 12 Commerce Mathematics Unit VI Probability : Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

### Download here CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes in pdf format

### CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes

CBSE Class 12 Commerce Mathematics Unit VI Probability : Probability can be expressed in terms such as impossible, unlikely, likely, or certain, as a number between 0 and 1, or as a percent between 0% and 100% as illustrated on the number line.

Probability | is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions |

Conditional probability | The probability that an event occurs given the outcome of some other event. Usually written, Pr(A l B). For example, the probability of a person being color blind given that the person is male is about 0.1, and the corresponding probability given that the person is female is approximately 0.0001. It is not, of course, necessary that Pr(A l B) = Pr(A l B); the probability of having spots given that a patient has measles, for example, is very high, the probability of measles given that a patient has spots is, however, much less. If Pr(A l B) = Pr(A l B) then the events A and B are said to be independent. |

Experimental probability | measures the likelihood that the event occurs based on the actual results of the experiment |

General addition rule | For any two events A and B |

General multiplication rule | The probability that both of two events A and B happen together can be found b |

Independent events | events for which the outcome of the second event does not depend on previous outcomes |

Dependent events | the occurrence of an event does affect how another event occurs |

Union | combine sets |

Complement | The set of all elements in the universal set that are not in that set. (A’) |

Factorials | n! is defined as the product of all the integers from 1 to n |

Subset | a set, all of whose members comprise part of a larger set. so (1, 2, 4) is a subset of (1, 2, 3, 4, 7, 9, 19) |

Mutually exclusive | events that have no outcomes in common |

Theoretical probability | = # of desired outcomes of the events / # of total outcomes |

Geometric probability | Probability that involves a geometric measure such as length or area |

Intersection | appears in both sets |

Fundamental counting principle | m x n = Number of ways that two events can occur, given that the first event can occur m ways and the second event can occur n ways |

Relative frequency | The relative frequency of a category or a numerical value is the associated frequency divided by the total number of data. Relative frequencies may be expressed in terms of percents, fractions, or decimals. A relative frequency distribution is a table or graph that presents the relative frequencies of the categories or numerical values |

### CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes

CBSE Class 12 Commerce Mathematics Unit VI Probability : Probability models can be helpful in predicting the probability of an event. Like shown to the right, all the possible outcomes of the toss of two coins can be represented as an organized list, table or tree diagram. The sample space becomes a model when a probability of each simple event , such as the chance of getting exactly two heads, is specified. This model shows that the theoretical probability of getting two heads is 1 out of 4 tosses or 25%. This may or may not happen in an actual experiment of many coin tosses, which is experimental probability.

An independent event is one that does not depend on another event. For example, the probability of picking a white marble from those shown to the right is 1 out of 7, 1/7, or about 14%. But what would be the chance of picking the white marble two times in a row? This is a compound event. The chance of white on the first draw is 1/7, and the chance of picking the white marble again is 1/7. The compound probability of picking white two times in a row is 1/7 x 1/7 or 1/49 which is about a 2% probability.

#### Approximate Instructional Time: 10 days

**Learning Intentions:**

- Understand, explain, and use
- Conditional Probabilities
- The Addition Rule for probabilities
- The Multiplication Rule for probabilities

**Standards:** NC.M2.S.IC.2 , NC.M2.S.CP.1 , NC.M2.S.CP.3 , NC.M2.S.CP.4 , NC.M2.S.CP.5 , NC.M2.S.CP.6 , NC.M2.S.CP.7 , NC.M2.S.CP.8

#### Making Inference and Justifying Conclusions Understand and evaluate random processes underlying statistical experiments.

NC.M2.S-IC.2 Use simulation to determine whether the experimental probability generated by sample data is consistent with the theoretical probability based on known information about the population.

Conditional Probability and the Rules for Probability

Understand independence and conditional probability and use them to interpret data.

NC.M2.S-CP.1 Describe events as subsets of the outcomes in a sample space using characteristics of the outcomes or as unions, intersections and complements of other events.

NC.M2.S-CP.3 Develop and understand independence and conditional probability.

NC.M2.S-CP.3a a. Use a 2-way table to develop understanding of the conditional probability of A given B (written P(A|B)) as the likelihood that A will occur given that B has occurred. That is, P(A|B) is the fraction of event B’s outcomes that also belong to event A.

NC.M2.S-CP.3b b. Understand that event A is independent from event B if the probability of event A does not change in response to the occurrence of event B. That is P(A|B)=P(A).

NC.M2.S-CP.4 Represent data on two categorical variables by constructing a two-way frequency table of data. Interpret the two-way table as a sample space to calculate conditional, joint and marginal probabilities. Use the table to decide if events are independent.

NC.M2.S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations

Conditional Probability and the Rules for Probability

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

NC.M2.S-CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in context.

NC.M2.S-CP.7 . Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in context.

NC.M2.S-CP.8 Apply the general Multiplication Rule P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in context. Include the case where A and B are independent: P(A and B) = P(A) P(B).

#### “I Can” Help My Student…

• Perform an experiment and collect data on a chance event, such as rolling die or picking names out of a hat.

• Explain why as probabilities approach 100% or 1, they become more likely.

• Use probability models to find probabilities of events.

• Compare theoretical and experimental probability.

• Represent probabilities of simple and compound events as a fraction, decimal, or percent.

• Create organized lists, tables, tree diagrams, and simulations to determine the probability of compound events.

Playing games is a wonderful way to practice skills at home in a fun environment. Stack-nPack books contain several math games covering math concepts from Kindergarten through High School. Stack-n-Pack card games may be checked out from your school (contact your school’s Parent Liaison) or purchased online: Stack-n-Pack Mathematics Card Games for KHS . The Variables game from the Stack-n-Pack 6-8 book is good practice for 7th graders.

### CBSE Class 12 Commerce Mathematics Unit VI Probability Complete Notes

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