Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is:  
Option: 1 \frac{2}{3}
Option: 2 \frac{8}{17}
Option: 3 \frac{4}{17 }
Option: 4 \frac{2}{5}

Answers (1)

\\\text{Let} \mathrm{B}_{1} \text{be the event where Box-I is selected and} \mathrm{ \;B}_{2} \text{ where box-II selected }\\\mathrm{P}\left(\mathrm{B}_{1}\right)=\mathrm{P}\left(\mathrm{B}_{2}\right)=\frac{1}{2}

Let E be the event where the selected card is non-prime.

For B1 : Prime numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

For B2 : Prime numbers: {31, 37, 41, 43, 47}

\begin{aligned} &\begin{aligned} \mathrm{P}(\mathrm{E}) &=\mathrm{P}\left(\mathrm{B}_{1}\right) \times \mathrm{P}\left(\frac{\mathrm{E}}{\mathrm{B}_{1}}\right)+\mathrm{P}\left(\mathrm{B}_{2}\right) \mathrm{P}\left(\frac{\mathrm{E}}{\mathrm{B}_{2}}\right) \\ &=\frac{1}{2} \times \frac{20}{30}+\frac{1}{2} \times \frac{15}{20} \end{aligned}\\ &\text { Required probability: }\\ &P\left(\frac{B_{1}}{E}\right)=\frac{\frac{1}{2} \times \frac{20}{30}}{\frac{1}{2} \times \frac{20}{30}+\frac{1}{2} \times \frac{15}{20}}=\frac{\frac{2}{3}}{\frac{2}{3}+\frac{3}{4}}=\frac{8}{17} \end{aligned}

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