A bag contains $5$ black, $7$ red, and $3$ white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is red, black or white and not a black one.

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Solution

Step 1: Finding the probability that the ball drawn is red:
According to the question, the total number of balls is $5 + 7 + 3 = 15$
The total number of red balls is $7$
We know that, $P r o b a b i l i t y = \frac{N u m b e r o f f a v o r a b l e o u t c o m e s}{T o t a l n u m b e r o f o u t c o m e s}$
Therefore,
$\Rightarrow P (r e d b a l l) = \frac{7}{15}$
Thus, the probability of drawing a red ball $= \frac{7}{15}$
Step 2: Finding the probability that the ball drawn is black or white:
According to the question,
The total number of balls is $5 + 7 + 3 = 15$
The total number of black or white balls is $5 + 3 = 8$
We know that, $P r o b a b i l i t y = \frac{N u m b e r o f f a v o r a b l e o u t c o m e s}{T o t a l n u m b e r o f o u t c o m e s}$
Therefore,
$\Rightarrow P (b l a c k o r w h i t e b a l l) = \frac{8}{15}$
Thus, the probability of drawing a white or black ball $= \frac{8}{15}$
Step 3: Finding the probability that the ball drawn is not black:
According to the question, the total number of balls is $5 + 7 + 3 = 15$
The total number of black balls is $5$
We know that, $P r o b a b i l i t y = \frac{N u m b e r o f f a v o r a b l e o u t c o m e s}{T o t a l n u m b e r o f o u t c o m e s}$
Therefore,
$\Rightarrow P (b l a c k b a l l) = \frac{5}{15} = \frac{1}{3}$
Thus, the probability of drawing a black ball $P (E) = \frac{5}{15} = \frac{1}{3}$
But, this is not what is required.
And, we know that the sum of the probability of occurrence of an event and the probability of non-occurrence of an event is $1$ .
$P (E) + P (E) = 1 \Rightarrow P (b l a c k) + P (n o t b l a c k) = 1 \Rightarrow \frac{1}{3} + P (E) = 1 \Rightarrow P (E) = 1 - \frac{1}{3} \Rightarrow P (E) = \frac{2}{3}$
Thus, the probability of drawing a ball that is not black is $\frac{2}{3} .$
Hence, from the conclusions made, the probability of drawing a red ball is $\frac{7}{15}$ , a white or black ball is $\frac{8}{15}$ , not a black ball is $\frac{2}{3}$

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