Question

A bag contains 5 white and 3 black balls, and 4 are successively drawn out and not replaced; what is the chance that they are alternately of different colors?

Answer 1

Here we will first find the probability of drawing white ball and then we will find the probability of drawing black balls. Then using these probabilities we will find the probability of getting alternating colors when the first ball is white and then we will find the Probability of getting alternating colors when the first ball is black. Then for resultant probabilities, we will find the sum of these two probabilities.

Complete step by step solution:
There are 5 white and 3 black balls.
Therefore, total number of total balls $=5+3=8$.
Now, we will find the probability of drawing white balls which is equal to the ratio of the number of white balls to the total number of balls.
Probability of drawing white ball $=\dfrac{5}{8}$
Now, we will find the probability of drawing black balls which is equal to the ratio of the number of black balls to the total number of balls.
Probability of drawing black ball $=\dfrac{3}{8}$
Now, we will find the probability of getting alternating colors when the first ball is white which will be equal to the alternative product of probabilities of white and black balls.
Therefore,
The probability of getting alternating colors when the first ball is white $=\dfrac{5}{8}\times \dfrac{3}{7}\times \dfrac{5}{6}\times \dfrac{3}{5}$
On multiplying the fractions, we get
The probability of getting alternating colors when the first ball is white $=\dfrac{1}{14}$
Now, we will find the probability of getting alternating colors when the first ball is black which will be equal to the alternative product of probabilities of white and black balls.
Therefore,
The probability of getting alternating colors when the first ball is white $=\dfrac{3}{8}\times \dfrac{5}{7}\times \dfrac{3}{6}\times \dfrac{5}{5}$
On multiplying the fractions, we get
The probability of getting alternating colors when the first ball is white $=\dfrac{1}{14}$

Thus, the resultant probability $=\dfrac{1}{14}+\dfrac{1}{14}=\dfrac{1}{7}$

Note:
Here have obtained the probability of getting alternating colors. Let’s define the term probability to understand it deeply. Here probability is defined as the ratio of number of favorable or required outcomes to the total number of outcomes. The probability can’t exceed the value of one.