Probability - Algebra - Mathematics Previous Year Questions

The probability distribution of $X$ is

$$\begin{array}{|c|l|c|c|c|} \hline \boldsymbol{X} & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{P}(\boldsymbol{X}) & 0.3 & k & 2 k & 2 k \\ \hline \end{array}$$

A man speaks truth 2 out of 3 times. He picks one of the natural numbers in the set \(S=\{1,2,3,4,5,6,7\}\) and reports that it is even. The probability that is actually even is

If \(A\) and \(B\) are two events of a sample space \(S\) such that \(P(A)=0.2, P(B)=0.6\) and \(P(A \mid B)=0.5\) then \(P\left(A^{\prime} \mid B\right)=\)

Two letters are chosen from the letters of the word 'EQUATIONS'. The probability that one is vowel and the other is consonant is

A random variable '\(X\)' has the following probability distribution

Then the value of \(k\) is

If '\(X\)' has a binomial distribution with parameters \(n=6, p\) and \(P(X=2)=12\), \(P(X=3)=5\) then \(P=\)

A die is thrown 10 times, the probability that an odd number will come up at least one time is

If \(A\) and \(B\) are two events such that \(P(A)=\frac{1}{3}, P(B)=\frac{1}{2}\) and \(P(A \cap B)=\frac{1}{6}\), then \(P\left(A^{\prime} / B\right)\) is

If \(A, B, C\) are three mutually exclusive and exhaustive events of an experiment such that \(P(A)=2 P(B)=3 P(C)\), then \(P(B)\) is equal to

The probability of solving a problem by three persons \(A, B\) and \(C\) independently is \(\frac{1}{2}, \frac{1}{4}\) and \(\frac{1}{3}\) respectively. Then the probability of the problem is solved by any two of them is

Events \(E_1\) and \(E_2\) from a partition of the sample space \(S\). \(A\) is any event such that \(P\left(E_1\right)=P\left(\dot{E}_2\right)=\frac{1}{2}, P\left(E_2 / A\right)=\frac{1}{2}\) and \(P\left(A / E_2\right)=\frac{2}{3}\), then \(P\left(E_1 / A\right)\) is

Given that, \(A\) and \(B\) are two events such that \(P(B)=\frac{3}{5}, P\left(\frac{A}{B}\right)=\frac{1}{2}\) and \(P(A \cup B)=\frac{4}{5}\), then \(P(A)\) is equal to

A car manufacturing factory has two plants \(X\) and \(Y\). Plant \(X\) manufactures \(70 \%\) of cars and plant \(Y\) manufactures \(30 \%\) of cars. \(80 \%\) of cars at plant \(X\) and \(90 %\) of cars at plant \(Y\) are rated as standard quality. A car is chosen at random and is found to be standard quality. The probability that it has come from plant \(X\) is :