Let S be a sample space of all 3X3 matrices with entries from the set {0,1}. Let the events E1 and E2 be given by E1={A∈S:detA=0} and E2={A∈S:sum of entries of A is 7} If a matrix is chosen at random from S , then the conditional probability 2P(E1/E2) equals----

n(S)=29=512E2 contains those matrices in which sum of entries is 7There will be 7 one’s , 2 zeroes n(E2)=9C2=36E1∩E2 contains those matrices in which 7 ones , 2 zeroesn(E1∩E2)=3C1×3C1×2=18P(E1/E2)=P(E1∩E2)P(E2)18/51236/512=12=0.50

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Let S be a sample space of all 3X3 matrices with entries from the set {0,1}. Let the events $E_{1} a n d E_{2}$ be given by $E_{1} = {A \in S : \det A = 0}$ and $E_{2} = {A \in S : s u m o f e n t r i e s o f A i s 7}$ If a matrix is chosen at random from S , then the conditional probability 2 $P (E_{1} / E_{2})$ equals----

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Detailed Solution

$n (S) = 2^{9} = 512$

$E_{2}$ contains those matrices in which sum of entries is 7

There will be 7 one’s , 2 zeroes $n (E_{2}) = 9_{C_{2}} = 36$

$E_{1} \cap E_{2}$ contains those matrices in which 7 ones , 2 zeroes

$n (E_{1} \cap E_{2}) = 3_{C_{1}} \times 3_{C_{1}} \times 2 = 18$

$P (E_{1} / E_{2}) = \frac{P (E_{1} \cap E_{2})}{P (E_{2})}$

$\frac{18 / 512}{36 / 512} = \frac{1}{2} = 0.50$

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