Question

A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?
$\text{A}\text{. }\dfrac{10}{21}$
$\text{B}\text{. }\dfrac{11}{21}$
$\text{C}\text{. }\dfrac{2}{7}$
$\text{D}\text{. }\dfrac{5}{7}$

Answer 1

Hint: First find total probabilities. Now check the number of possibilities favorable to the given event. The ratio of these two will become the probability of the given event happening. This probability is the required result. $\text{Probability=}\dfrac{\text{Favorable cases}}{\text{Total cases}}$

Complete step-by-step answer:
Given number of balls in bag with its color are:
2 red, 3 green and 2 blue.
Given number of balls which are drawn at random as:
2 balls are drawn.
Condition on the above drawn 2 balls, is given by:
None of them must be blue.
Meaning of above condition can be said in other way as:
The 2 balls drawn must be either red or green.
Total balls in the bag are given by as follows:
Total balls = sum of all color of balls.
By substituting their values, we can say the total as:
Total balls $=2+3+2$ balls.
By simplifying the above equation we can say it as:
Total $=7$ balls
The process of drawing the balls is nothing but selecting of balls, the concept of combination is analogous
So, the total number of ways possible to select 2 balls front:
${}^{7}{{C}_{2}}=\text{total ways}$
By applying formula for above equation, we can write it as:
$\text{total ways =}\dfrac{7!}{5!2!}=\dfrac{7\times 6}{2}$
By simplifying the above equation, we can write the value as:
$\text{total ways=}\dfrac{42}{2}=21$
Favorable colors for drawing are red balls, green balls.
So, favorable balls number can be given as follows:
Favorable balls $=2\text{ red +3 green}$
Favorable cases are calculated by the sum above, given as:
5 balls.
To select 2 balls from these, we can write it as:
${}^{5}{{C}_{2}}$
By substituting the formula, we can write its value to be:
Favorable $=\dfrac{5!}{2!3!}=\dfrac{5\times 4}{2}$
By simplifying the above equation we can get its value to be:
Favorable $=\dfrac{20}{2}=10$
Probability:- in a simple way, probability is how likely something is to happen. Whenever we’re unsure about the outcome of an event we talk about probabilities. It is the value which lies between 0 and 1. Probability is the division result of favorable outcomes, total outcomes.
$\text{Probability=}\dfrac{\text{Favorable cases}}{\text{Total cases}}$
By substituting the values, we can write its value as:
$\text{probability=}\dfrac{10}{21}$
Therefore, given (a) is the correct answer.

Note: Be careful while taking total cases, because if you miss any one case also the whole probability might change. Alternate is checking the probability for at least 1 blue and subtract the probability from 1 to get the result. Anyways you will get the same answer because both of them define the same event.