Let S be the sample space of all 3 $$ \times $$ 3 matrices with entries from the set {0, 1}. Let the events E₁ and E₂ be given by

E₁ = {A$$ \in $$S : det A = 0} and

E₂ = {A$$ \in $$S : sum of entries of A is 7}.

If a matrix is chosen at random from S, then the conditional probability P(E₁ | E₂) equals ...............

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least $$0.96,$$ is

Of the three independent events $${E_1},{E_2}$$ and $${E_3},$$ the probability that only $${E_1}$$ occurs is $$\alpha ,$$ only $${E_2}$$ occurs is $$\beta $$ and only $${E_3}$$ occurs is $$\gamma .$$ Let the probability $$p$$ that none of events $${E_1},{E_2}$$ or $${E_3}$$ occurs satisfy the equations $$\left( {\alpha -2\beta } \right)p = \alpha \beta $$ and $$\left( {\beta - 3\gamma } \right)p = 2\beta \gamma .$$ All the given probabilities are assumed to lie in the interval $$(0, 1)$$.

Then $${{\Pr obability\,\,of\,\,occurrence\,\,of\,\,{E_1}} \over {\Pr obability\,\,of\,\,occurrence\,\,of\,\,{E_3}}}$$