A recent college graduate is planning to take the first three actuarial examinations in the coming summer. She will take the first actuarial exam in June. If she passes that exam, then she will take the second exam in July, and if she also passes that one, then she will take the third exam in September. If she fails an exam, then she is not allowed to take any others. The probability that she passes the first exam is 0.3. If she passes the first exam, then the conditional probability that she passes the second one is 0.5, and if she passes both the first and the second exams, then the conditional probability that she passes the third exam is 0.6. a) What is the probability that she does not pass all three exams? (Hint: Use the multiplication rule.) b) Given that she did not pass all three exams, what is the conditional probability that she passed the first and the second exams? c) Given that she did not pass all three exams, what is the conditional probability that she failed the first exam?
Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
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Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
A recent college graduate is planning to take the first three actuarial examinations in the coming summer. She will take the first actuarial exam in June. If she passes that exam, then she will take the second exam in July, and if she also passes that one, then she will take the third exam in September. If she fails an exam, then she is not allowed to take any others. The
a) What is the probability that she does not pass all three exams? (Hint: Use the multiplication rule.)
b) Given that she did not pass all three exams, what is the conditional probability that she passed the first and the second exams?
c) Given that she did not pass all three exams, what is the conditional probability that she failed the first exam?
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