a)
To explain: Whether the events A, and B are mutually exclusive or not and also verify if they are independent on not.
Events A and B are not mutually exclusive events and they are independent events.
Given:The set of numbers
Event A is selecting a multiple of 2 and event B is selecting a multiple of 3.
Concept used:
Two events A and B are said to be mutually exclusive when they do not occur simultaneously and
Calculation:
With respect to the given events, the probability of event A is calculated as shown below
The numbers in the set which are multiples of 2 are
The total number of numbers in the set
The probability that randomly selected number is a multiple of 2 is
Similarly the probability that event B is calculated as shown below
The numbers in the set which are multiples of 3 are
The total number of numbers in the set
The probability that a randomly selected number is a multiple of 3 is
The numbers which are multiples of both 2 and 3 are
The probability that a randomly selected number is a multiple of both 2 and 3 is
This satisfy the identity
Conclusion:The events A and B can occur simultaneously and
b)
To calculate: The probabilities of occurrence of events A and B.
The probabilities of occurrence of events A and B are
Concept used:
The probabilities of a particular event is the ratio of number of favourable events to the total possible outcomes.
Calculation:With respect to the given events, the probability of event A is calculated as shown below
The numbers in the set which are multiples of 2 are
The total number of numbers in the set
The probability that a randomly selected number is a multiple of 2 is
Similarly, the probability that event B is calculated as shown below
The numbers in the set which are multiples of 3 are
The total number of numbers in the set
The probability that a randomly selected number is a multiple of 3 is
Conclusion:The probabilities of the occurrence of the events A and B are
c)
To calculate: The probability that both events A and B occur together.
The probability that both events A and B occur together is
Concept used:
Two events A and B are said to be independent events when the occurrence of one event does not impact the occurrence of the other event and
Calculation:
The given events A and B are independent events and its probability is calculated as shown below
It can also be calculated as shown below
The numbers which are multiples of both 2 and 3 are
The probability that a randomly selected number is a multiple of both 2 and 3 is
Conclusion:The probability that events A and B occur simultaneously is
d)
To calculate: The probability that both events A or B occur
The probability that both events A or B occur
Formula used:
The probability that events A or B is calculated using the formula
Calculation:
The probability that events A or B occur is calculated as shown below
Conclusion:The probability that both events A or B occur
e)
To calculate: The conditional probabilities
The conditional probabilities
Formula used:
The conditional probabilities are calculated using the formula
Calculation:
The conditional probabilities are calculated as shown below
Conclusion:The events A and B are independent so the occurrence of one event has no impact on other event. The conditional probabilities are
Chapter 11 Solutions
High School Math 2015 Common Core Algebra 2 Student Edition Grades 10/11
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