Q8(ii):
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 (ii) not red?

Solution :

Given:

Number of red balls in the bag ($n_R$) = $3$

Number of black balls in the bag ($n_B$) = $5$

To Find:

The probability that the ball drawn at random is not red, denoted as $P(\text{not Red})$.

Step 1: Calculate the total number of possible outcomes.

The total number of outcomes is the sum of all balls present in the bag.

Total number of balls ($n(S)$) = $n_R + n_B$

$n(S) = 3 + 5 = 8$

[Since the bag contains only red and black balls, the total sample space consists of 8 equally likely outcomes.]

Step 2: Identify the favorable outcomes for the event "not red".

A ball that is "not red" must be a black ball in this specific context.

Number of favorable outcomes ($n(E)$) = Number of black balls = $5$

Step 3: Apply the probability formula.

The probability of an event $E$ is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

$P(\text{not Red}) = \frac{n(E)}{n(S)}$

$P(\text{not Red}) = \frac{5}{8}$

Alternative Method (Using Complementary Events):

We know that the sum of the probability of an event and its complement is 1: $P(E) + P(\text{not } E) = 1$.

First, calculate the probability of drawing a red ball:

$P(\text{Red}) = \frac{n_R}{n(S)} = \frac{3}{8}$

Now, calculate the probability of not drawing a red ball:

$P(\text{not Red}) = 1 - P(\text{Red})$

$P(\text{not Red}) = 1 - \frac{3}{8}$

$P(\text{not Red}) = \frac{8 - 3}{8} = \frac{5}{8}$

Final Answer: The probability that the ball drawn is not red is $\frac{5}{8}$.

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