Answer
Verified
421.8k+ views
Hint: Let us denote P be the event that the student passes in physics and let us denote the M be the event that the student passes in mathematics. The probability that the student passes in mathematics given that he passes in physics is given by the formula $P\left( M|P \right)=\dfrac{P\left( M\cap P \right)}{P\left( P \right)}$. Using this formula, we can solve this question.
Complete step by step solution:
Before proceeding with the question, we must know all the formulas that will be required to solve this question.
If we are given two events A and B, then the probability of event A given that event B will also occur is given by the formula,
$P\left( A|B \right)=\dfrac{P\left( A\cap B \right)}{P\left( B \right)}$ . . . . . . . . . . . . . (1)
Also, in probability, we have a formula $P\left( A\cap B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cup B \right)$ . . . . . . . . . (2)
For this question, let us denote P be the event that the student passes in physics and let us denote the M be the event that the student passes in mathematics.
It is given that 70 % students pass in physics, so, $P\left( P \right)=\dfrac{70}{100}$.
$\Rightarrow P\left( M \right)=\dfrac{7}{10}$
It is given that 75 % students pass in mathematics, so, $P\left( M \right)=\dfrac{75}{100}$.
$\Rightarrow P\left( P \right)=\dfrac{3}{4}$
Also, it is given that 10 % of the students fail in both subjects. So, we can say 90% of the students passed either in physics, or in mathematics, or in both.
$\begin{align}
& \Rightarrow P\left( M\cup P \right)=\dfrac{90}{100} \\
& \Rightarrow P\left( M\cup P \right)=\dfrac{9}{10} \\
\end{align}$
Using formula (2), we get,
\[\begin{align}
& P\left( M\cap P \right)=\dfrac{7}{10}+\dfrac{3}{4}-\dfrac{9}{10} \\
& \Rightarrow P\left( M\cap P \right)=\dfrac{28+30-36}{40} \\
& \Rightarrow P\left( M\cap P \right)=\dfrac{22}{40} \\
& \Rightarrow P\left( M\cap P \right)=\dfrac{11}{20} \\
\end{align}\]
Using formula (1), the probability that the student passes in mathematics given that he passes in physics is equal to,
$\begin{align}
& P\left( M|P \right)=\dfrac{P\left( M\cap P \right)}{P\left( P \right)} \\
& \Rightarrow P\left( M|P \right)=\dfrac{\dfrac{11}{20}}{\dfrac{7}{10}} \\
& \Rightarrow P\left( M|P \right)=\dfrac{11}{14} \\
\end{align}$
Hence, the answer is $\dfrac{11}{14}$.
Note: There is a possibility that one may commit a mistake while using the formula $P\left( M|P \right)=\dfrac{P\left( M\cap P \right)}{P\left( P \right)}$. Instead of the correct formula $P\left( M|P \right)=\dfrac{P\left( M\cap P \right)}{P\left( P \right)}$, it is possible that one may use the formula $P\left( M|P \right)=\dfrac{P\left( M\cap P \right)}{P\left( M \right)}$ which will lead us to an incorrect answer.
Complete step by step solution:
Before proceeding with the question, we must know all the formulas that will be required to solve this question.
If we are given two events A and B, then the probability of event A given that event B will also occur is given by the formula,
$P\left( A|B \right)=\dfrac{P\left( A\cap B \right)}{P\left( B \right)}$ . . . . . . . . . . . . . (1)
Also, in probability, we have a formula $P\left( A\cap B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cup B \right)$ . . . . . . . . . (2)
For this question, let us denote P be the event that the student passes in physics and let us denote the M be the event that the student passes in mathematics.
It is given that 70 % students pass in physics, so, $P\left( P \right)=\dfrac{70}{100}$.
$\Rightarrow P\left( M \right)=\dfrac{7}{10}$
It is given that 75 % students pass in mathematics, so, $P\left( M \right)=\dfrac{75}{100}$.
$\Rightarrow P\left( P \right)=\dfrac{3}{4}$
Also, it is given that 10 % of the students fail in both subjects. So, we can say 90% of the students passed either in physics, or in mathematics, or in both.
$\begin{align}
& \Rightarrow P\left( M\cup P \right)=\dfrac{90}{100} \\
& \Rightarrow P\left( M\cup P \right)=\dfrac{9}{10} \\
\end{align}$
Using formula (2), we get,
\[\begin{align}
& P\left( M\cap P \right)=\dfrac{7}{10}+\dfrac{3}{4}-\dfrac{9}{10} \\
& \Rightarrow P\left( M\cap P \right)=\dfrac{28+30-36}{40} \\
& \Rightarrow P\left( M\cap P \right)=\dfrac{22}{40} \\
& \Rightarrow P\left( M\cap P \right)=\dfrac{11}{20} \\
\end{align}\]
Using formula (1), the probability that the student passes in mathematics given that he passes in physics is equal to,
$\begin{align}
& P\left( M|P \right)=\dfrac{P\left( M\cap P \right)}{P\left( P \right)} \\
& \Rightarrow P\left( M|P \right)=\dfrac{\dfrac{11}{20}}{\dfrac{7}{10}} \\
& \Rightarrow P\left( M|P \right)=\dfrac{11}{14} \\
\end{align}$
Hence, the answer is $\dfrac{11}{14}$.
Note: There is a possibility that one may commit a mistake while using the formula $P\left( M|P \right)=\dfrac{P\left( M\cap P \right)}{P\left( P \right)}$. Instead of the correct formula $P\left( M|P \right)=\dfrac{P\left( M\cap P \right)}{P\left( P \right)}$, it is possible that one may use the formula $P\left( M|P \right)=\dfrac{P\left( M\cap P \right)}{P\left( M \right)}$ which will lead us to an incorrect answer.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Select the word that is correctly spelled a Twelveth class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What organs are located on the left side of your body class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE