Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red?

A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red?

Answers (1)

Solution

Given

A bag contains 5 red marbles and 3 black marbles

If the first marble is red, the following conditions have to be followed for at least one marble to be black.

  1. The second marble is black and the third is red = E_1
  2.  Second and third, both marbles are black = E_2

Second marble is red and the third marble is black = E_3
Let event R_n = drawing red marble in n^{th} draw
And event R_n = drawing black marble in n^{th} draw

\\\therefore P(E_ 1)=P\left(R_{1}\right) \cdot P\left(B_{1} / R_{1}\right) \cdot P\left(R_{2} / B_{1} R_{1}\right)=\frac{5}{8} \times \frac{3}{7} \times \frac{4}{6}=\frac{5}{28} \\\\\therefore \mathrm{P}\left(\mathrm{E}_{2}\right)=\mathrm{P}\left(\mathrm{R}_{1}\right) \cdot \mathrm{P}\left(\mathrm{B}_{1} / \mathrm{R}_{1}\right) \cdot \mathrm{P}\left(\mathrm{B}_{2} / \mathrm{B}_{1} \mathrm{R}_{1}\right)=\frac{5}{8} \times \frac{3}{7} \times \frac{2}{6}=\frac{5}{56} \\\\\therefore \mathrm{P}\left(\mathrm{E}_{3}\right)=\mathrm{P}\left(\mathrm{R}_{1}\right) \cdot \mathrm{P}\left(\mathrm{R}_{2} / \mathrm{R}_{1}\right) \cdot \mathrm{P}\left(\mathrm{B}_{1} / \mathrm{R}_{1} \mathrm{R}_{2}\right)=\frac{5}{8} \times \frac{4}{7} \times \frac{3}{6}=\frac{5}{28}    
Hence, for the probability for at least one marble to be black is P

(\mathrm{E})=\mathrm{P}\left(\mathrm{E}_{1}\right)+\mathrm{P}\left(\mathrm{E}_{2}\right)+\mathrm{P}\left(\mathrm{E}_{3}\right)\\ Therefore, P(E)=\frac{5}{28}+\frac{5}{56}+\frac{5}{28}=\frac{25}{56}

Posted by

infoexpert22

View full answer

Similar Questions

Latest Question