In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing at least one of them is 0.95. What is the probability of passing both?
Solution and Explanation
Let A and B be the events of passing first and second examinations respectively.
Accordingly, P(A) = 0.8, P(B) = 0.7 and P (A or B) = 0.95
We know that P (A or B) = P(A) + P(B) - P (A and B)
∴ 0.95 = 0.8 + 0.7 - P (A and B)
\(⇒\)P (A and B) = 0.8 + 0.7 - 0.95 = 0.55
Thus, the probability of passing both the examinations is 0.55.
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Concepts Used:
Axiomatic Approach to Probability
The axiomatic probability perspective is a unifying perspective in which the coherent conditions used in theoretical and experimental probability exhibit subjective probability. Kolmogorov's set of rules or axioms is put to all types of probability. They are known as Kolmogorov's 3 axioms by mathematicians. You can use axiomatic probability to calculate the likelihood of an event that is occurring or not occurring.
The 3 axioms are applicable to all other probability perspectives. This viewpoint is defined as the probability of any function from numbers to events that are satisfied by the three axioms listed below:
- The greatest possible probability is one, and the least possible probability is zero.
- A determined event has a probability of one.
- Two mutually exclusive events cannot happen at the same time, but the union of events states that any one of them can.
