Question

An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?

Answer 1

Hint – In this particular type of question use the concept that we have to find the probability without replacement so after drawing first ball (say black color), next time when we draw the ball the number of balls of same color are decreased by 1 and so for the total number of balls so probability is changed accordingly, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
In an urn there are 10 black balls and 5 white balls.
So the total number of balls = (10 + 5) = 15 balls.
Now two balls are drawn from the urn one after the other without replacement and we have to find the probability that both drawn balls are black.
So we have to find the probability without replacement so after drawing the first ball (say black color), next time when we draw the ball the number of balls of the same color are decreased by 1 and so for the total number of balls so probability is changed accordingly.
Now as we know that the probability (P) is the ratio of the favorable number of outcomes to the total number of outcomes so we have,
$ \Rightarrow P = \dfrac{{{\text{favorable number of outcomes}}}}{{{\text{total number of outcomes}}}}$
So when we draw first ball the favorable number of outcomes = 10 (black balls)
And the total number of outcomes = 15.
So the probability of randomly chosen first black ball is, ${P_1} = \dfrac{{{\text{10}}}}{{{\text{15}}}}$
Now from the urn one black ball is decreased (as the balls are chosen without replacement) so the remaining lack balls = (10 – 1) = 9,
And the total number of balls are also decreased by one so the total number of balls are (15 – 1) = 14
Now when the second ball is drawn.
So the probability of randomly chosen second black ball is, ${P_2} = \dfrac{9}{{{\text{14}}}}$
So the probability (P) that both drawn balls are black = ${P_1}{P_2}$
$ \Rightarrow P = \dfrac{{10}}{{15}} \times \dfrac{9}{{14}} = \dfrac{3}{7}$
So this is the required answer.

Note – Whenever we face such types of question the key concept we have to remember is that always recall that the probability is the ratio of the favorable number of outcomes to the total number of outcomes, so first calculate these outcomes as above then put these outcomes in the formula of the probability and simplify and the total probability is the multiplication of the probability of first time chosen ball and the probability of second time chosen ball, we will get the required probability.