Question

An urn contains 6 red marbles and 4 black marbles. Two marbles are randomly drawn (one by one) from the urn without replacement. What is the probability that both marbles are black?

Answer 1

Answer:   The required probability is $\dfrac{2}{15}.$

Step-by-step explanation:  Given that an  urn contains 6 red marbles and 4 black marbles. Two marbles are randomly drawn one by one from the urn without replacement.

We are to find the probability that both drawn marbles are black.

Let E and F denote the events of two marbles one by one without replacement and let S and S' denote the corresponding sample spaces.

Then, we have

$n(E)=^4C_1=4,\\\\n(F)=^3C_1=3,\\\\n(S)=^{10}C_1=10,\\\\n(S')=^9C_1=9.$

Therefore, the probability  that both marbles are red is given by

$p=P(E)\times P(F)=\dfrac{n(E)}{n(S)}\times\dfrac{n(F)}{n(S')}=\dfrac{4}{10}\times\dfrac{3}{9}=\dfrac{2}{15}.$

Thus, the required probability is $\dfrac{2}{15}.$