Question

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is: -
(a) Red
(ii) Black

Answer 1

Hint: Assume n (R) as the number of red balls and n (B) as the number of black balls. To find the total number of balls, take the sum of the number of red balls and black balls and denote it with n (S). Now, to find the probability of drawing a red ball take the ratio of n (R) to n (S). Similarly, to find the probability of drawing black ball take the ratio of n (B) to n (S).

Complete step by step answer:
Let us assume the number of red balls, black balls in the bag are denoted by n (R), n (B) respectively. Also, we are assuming the total number of balls in the bag as n (S).
Now, it is given that there are 3 red balls and 5 black balls in the bag. Therefore, we have,
\[\Rightarrow \] n (R) = 3 and n (B) = 5
So, total number of balls in the bag will be the sum of the number of red balls and black balls, therefore,
\[\Rightarrow \] n (S) = n (R) + n (B)
\[\Rightarrow \] n (S) = 3 + 5
\[\Rightarrow \] n (S) = 8
Now, we have to determine the probability of drawing a red ball and a black ball if one ball is drawn at random from the bag.
We know that probability is the ratio of the number of favourable outcomes to the total number of outcomes. So, let us consider the two parts of the question one – by – one.
(i) Here, we have to find the probability of drawing a red ball. So, the required probability will be the ratio of the number of red balls to the total number of balls in the bag.
\[\Rightarrow \] P (R) = Probability of drawing a red ball = \[\dfrac{n(R)}{n(S)}\]
Substituting the value of n (R) and n (S), we get,
\[\Rightarrow \] P (R) = \[\dfrac{3}{8}\]
(ii) Here, we have to find the probability of drawing a black ball. So, the required probability will be the ratio of the number of black balls to the total number of balls in the bag.
\[\Rightarrow \] P (B) = Probability of drawing a black ball = \[\dfrac{n(B)}{n(S)}\]
Substituting the value of n (B) and n (S), we get,
\[\Rightarrow \] P (B) = \[\dfrac{5}{8}\]

Hence, probability of drawing a red ball and a black ball are \[\dfrac{3}{8}\] and \[\dfrac{5}{8}\] respectively.

Note: One may note an important result, that is P (R) + P (B) = 1. That means the sum of all the probabilities is equal to 1. So, we can determine the probability of drawing a black ball without using the formula \[\dfrac{n(B)}{n(S)}\]. What we will do is, we will find the probability of drawing a red ball by using the formula \[\dfrac{n(R)}{n(S)}\] and then subtract it from 1. One must remember this important theorem as it helps in reducing calculations in difficult probability problems.