Question

One bag contains 3 white balls, 7 red balls and 15 black balls. Another bag contains 10 white balls, 6 red balls and 9 black balls. One ball is taken from each bag. What is the probability that both the balls will be of the same color?
A. $\dfrac{{207}}{{625}}$
B. $\dfrac{{191}}{{625}}$
C. $\dfrac{{23}}{{625}}$
D. $\dfrac{{227}}{{625}}$

Answer 1

Here, we will find the probability that both the balls are of the same color. We will be using the probability formula to find the probability that both the balls are white, both the balls are red and both the balls are black. Then by adding all the probabilities, we will find the total probability.
Formula Used: The probability is given by the formula $P\left( A \right) = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}$ where $n\left( A \right)$ is the number of favorable outcomes and $n\left( S \right)$ is the total number of outcomes.

Complete Step by step Solution:
We are given that a bag contains 3 white balls, 7 red balls and 15 black balls. Consider the Bag as Bag A.
We are also given that another bag contains 10 white balls, 6 red balls and 9 black balls. Consider the Bag as Bag B.
We will find the total number of balls in the bags by adding the number of white balls, number of red balls and number of black balls in each bag.
Total Number of Balls in Bag A $= 3 + 7 + 15$
Adding the terms, we get
$\Rightarrow$ Total Number of Balls in Bag A $= 25$
So, the Sample Space of Bag A, $n\left( {{S_A}} \right) = 25$
Total Number of Balls in Bag B$= 10 + 6 + 9$
$\Rightarrow$ Total Number of Balls in Bag B$= 25$
So, the Sample Space of Bag B, $n\left( {{S_B}} \right) = 25$
We are given that two balls are taken from each bag at a time such that one ball from Bag A and another ball from Bag B and both the balls are of the same color.
Let W be the event of selecting a white ball from Bag A and a white ball from Bag B and $P\left( W \right)$ be the probability of selecting a white ball from Bag A and a white ball from Bag B.
The probability is given by the formula $P\left( A \right) = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}$
Now we will find the probability of selecting white balls. Therefore
$P\left( W \right) = \dfrac{{n\left( {{W_A}} \right)}}{{n\left( {{S_A}} \right)}} \cdot \dfrac{{n\left( {{W_B}} \right)}}{{n\left( {{S_B}} \right)}}$
$\Rightarrow P\left( W \right) = \dfrac{3}{{25}} \cdot \dfrac{{10}}{{25}}$
Multiplying the terms, we get
$\Rightarrow P\left( W \right) = \dfrac{{30}}{{{{\left( {25} \right)}^2}}}$
Let R be the event of selecting a red ball from Bag A and a red ball from Bag B and $P\left( R \right)$ be the probability of selecting a red ball from Bag A and a red ball from Bag B.
Now we will find the probability of selecting red balls. Therefore
$P\left( R \right) = \dfrac{{n\left( {{R_A}} \right)}}{{n\left( {{S_A}} \right)}} \cdot \dfrac{{n\left( {{R_B}} \right)}}{{n\left( {{S_B}} \right)}}$
$\Rightarrow P\left( R \right) = \dfrac{7}{{25}} \cdot \dfrac{6}{{25}}$
Multiplying the terms, we get
$\Rightarrow P\left( R \right) = \dfrac{{42}}{{{{\left( {25} \right)}^2}}}$
Let B be the event of selecting a black ball from Bag A and a black ball from Bag B and $P\left( B \right)$ be the probability of selecting a black ball from Bag A and a black ball from Bag B.
Now we will find the probability of selecting black balls. Therefore

$P\left( B \right) = \dfrac{{n\left( {{B_A}} \right)}}{{n\left( {{S_A}} \right)}} \cdot \dfrac{{n\left( {{B_B}} \right)}}{{n\left( {{S_B}} \right)}}$
$\Rightarrow P\left( B \right) = \dfrac{{15}}{{25}} \cdot \dfrac{9}{{25}}$
Multiplying the terms, we get
$\Rightarrow P\left( B \right) = \dfrac{{135}}{{{{\left( {25} \right)}^2}}}$
Now, we will find the probability that both the balls will be of the same color by adding the probabilities that both the balls are of same color.
Therefore, the total probability will be
$P = P\left( W \right) + P\left( R \right) + P\left( B \right)$
$\Rightarrow P = \dfrac{{30}}{{625}} + \dfrac{{42}}{{625}} + \dfrac{{135}}{{625}}$
Adding the terms, we get
$\Rightarrow P = \dfrac{{207}}{{625}}$
Therefore, the probability that both the balls will be of the same color is
$\dfrac{{207}}{{625}}$ .

Thus Option (A) is the correct answer.

Note:
We should remember that the bag contains three different colors of balls so the probability of two balls of the same color in two different bags has to be multiplied to find the probability that the two balls selected at random is of same color. Probability is the possibility of an event to occur. We will not use the formula of combination since both the balls are identical.