Question

In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first exam is \[0.8\] and the probability of passing the second exam is \[0.7\]. The probability of passing at least one of the exams is \[0.95\].What is the probability of passing both?

Answer 1

Hint: Probability of any event is represented as \[P\left( {event} \right)\]. If \[A\] and \[B\] be the two events then \[P\left( {A \cup B} \right)\] represents the probability of getting either \[A\] or \[B\] at a time and \[P(A \cap B)\] represents the probability of getting both \[A\] and\[B\]. Addition rule of probability establishes the relation between them as \[P\left( {A \cap B} \right) = P\left( A \right){\text{ }} + {\text{ }}P\left( B \right){\text{ }}-{\text{ }}P\left( {A \cup B} \right)\].

Complete step by step answer:
We have given the probability of a random chosen student passing the first exam, second exam and passing at least one exam. We have to find the probability of passing both the exams. Let \[A\] be the event of passing the first exam by a random chosen student, \[B\] be the event of passing the second exam. Then,
Probability of passing the first exam is given by, \[P\left( A \right) = 0.8\]
Probability of passing the first exam is given by, \[P\left( B \right) = 0.7\]

Probability of passing at least one exam, that is passing either of exams is given by \[P\left( {A \cup B} \right) = 0.95\]
Probability of passing both the exams is given by, \[P\left( {A \cap B} \right) = ?\]
Now from addition rule of probability we have,
\[P\left( {A \cap B} \right) = P\left( A \right){\text{ }} + {\text{ }}P\left( B \right){\text{ }}-{\text{ }}P\left( {A \cup B} \right)\]
Substituting the given values we have,
\[P\left( {A \cap B} \right) = 0.8 + 0.7 - 0.95\]
On simplifying we get
\[P\left( {A \cap B} \right) = 1.5 - 0.95\]
On subtracting we get,
\[\therefore P\left( {A \cap B} \right) = 0.55\]

Hence, Probability of passing both exams by a randomly chosen student is \[0.55\].

Note:Probability of any event cannot be less than \[0\] or greater than \[1\], it lies in between \[0\] and \[1\]. Probability has no unit as it is a ratio. Probability of getting \[A\] and \[B\] together for two independent variables \[A\] and \[B\] is given as \[P\left( {A \cap B} \right) = P\left( A \right) \times P\left( B \right)\]. In probability ‘or’ is denoted as Union and ‘and’ is denoted as intersection. The addition rule is derived from set theory.