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 bag balls will be the same colour?
a. $\dfrac{207}{625}$
b. $\dfrac{191}{625}$
c. $\dfrac{23}{625}$
d. $\dfrac{227}{625}$

Answer 1

Hint: If A and B are two independent events then probability that both the events occur simultaneously is equal to P(A).P(B) , where, P(A)=probability that event A is occurred and P(B)=probability that event B occurs.
Here the two independent equations are picking the balls of the same colours.

Complete step-by-step answer:
Given 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. If one ball is taken from each bag then we have to find the probability that the both balls will be of the same colour.
First, we will find the probability that the both balls taken from them will be of white colour and then do the same for both red and black balls.
We know that the probability that the both balls will be of white colour is the product of probabilities of picking the white ball from both bags because both are independent and simultaneous events.
Now, in bag 1 , total number of balls = 25.
Number of white balls = 3
So, the probability that we took white ball from bag 1 is $\dfrac{3}{25}$ .
In bag 2, total number of balls = 25,
Number of white balls = 10.
So, the probability that we took white ball from bag 2 is $\dfrac{10}{25}$
Therefore, the probability that both the balls will be of white colour is $\dfrac{3}{25}\times \dfrac{10}{25}$ .
$=\dfrac{30}{625}$

Now, we will consider the case of red balls. The number of red balls in bag 1 = 7.
So, the probability that we took the red ball from bag 1 is$\dfrac{7}{25}$.
The number of red balls in bag 2 = 6.
So, the probability that we took the red ball from bag 2 is $\dfrac{6}{25}$.
Therefore, probability that both the balls will be of red colour is $\dfrac{7}{25}\times \dfrac{6}{25}=\dfrac{42}{625}$.
Now, we will consider the case of black balls. In bag 1, the number of black balls = 15.
So, the probability that we took black ball from bag 1 is $\dfrac{15}{25}$.
In bag 2, the number of black balls = 9.
So, the probability that we took black ball from bag 2 is $\dfrac{9}{25}$.
Therefore, probability that both the balls will be of black colour is $\dfrac{15}{25}\times \dfrac{9}{25}=\dfrac{135}{625}$.
Probability that we took the same colour from both bags is a summation of the probability that we took red ball, white ball and black ball from both the bags simultaneously because if any one of the mentioned events occurs then our required event occurs.
Therefore, probability that the both balls will be of same colour is $\left( \dfrac{30}{625}+\dfrac{42}{625}+\dfrac{135}{625} \right)=\dfrac{207}{625}.$
Hence, the probability that the both balls will be of the same colour is $\dfrac{207}{625}$ .
So, the correct option is option A.

Note: While doing calculations, make sure that sign mistakes are not present. Sign mistakes are very common and can result in wrong answers. So, the students must be very careful while doing calculations.