• # question_answer A pack of playing cards was found to contain only 51 cards. If the first 13 cards, which are examined are all red, the probability that the missing card is black is A)  $\frac{2}{3}$                                   B)  $\frac{1}{3}$ C)  $\frac{2}{9}$                                   D)  None of these

Probability of getting 13 red cards is P (R) = last card is black than red card is selected + lost card is red than red card is selected $P(R)=P(LB)\cdot P\left( \frac{P}{LB} \right)+P(LR)P\left( \frac{R}{LR} \right)$ $P(R)=\frac{26}{52}\cdot \frac{^{26}{{C}_{13}}}{^{51}{{C}_{3}}}+\frac{26}{52}\cdot \frac{^{25}{{C}_{13}}}{^{51}{{C}_{13}}}$ $P(R)=\frac{1}{2}\cdot \frac{^{26}{{C}_{13}}}{^{51}{{C}_{3}}}+\frac{1}{2}\cdot \frac{^{25}{{C}_{13}}}{^{51}{{C}_{13}}}$ $P\left( \frac{LB}{R} \right)=\frac{P(LB).P\left( \frac{R}{LB} \right)}{P(R)}$ $=\frac{\frac{1}{2}\cdot \frac{^{26}{{C}_{13}}}{^{51}{{C}_{13}}}}{\frac{1}{2}.\frac{^{26}{{C}_{13}}}{^{51}{{C}_{13}}}+\frac{1}{3}\cdot \frac{^{25}{{C}_{13}}}{^{51}{{C}_{13}}}}=\frac{2}{3}$