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Question: A bag contains \(6\) red balls and some blue balls. If the probability of drawing a blue ball is twi...

A bag contains 66 red balls and some blue balls. If the probability of drawing a blue ball is twice that of a red ball, find the number of balls in the bag.

Explanation

Solution

Assume the number of blue balls as a variable and then find the probability of drawing the red ball and the probability of drawing the blue ball. Use the given condition to approach the required result.

Complete step by step solution:
It is given in the problem that there are 66 red balls and some blue balls in the bag and the probability of drawing a blue ball is twice that of a red ball.
The goal of the problem is to find the total number of balls in the bag.

Now, assume that there are xx blue balls in the bag. Then, we have
Number of red balls=n(R)=6 = n\left( R \right) = 6
Number of blue balls=n(B)=x = n\left( B \right) = x
So, the total number of balls in the bag is the sum of the number of blue balls and the number of red balls. That is,

Total number of balls in the bag =n(T)=6+x = n\left( T \right) = 6 + x
We know that the probability of any event, P(Event)=No. of favourable outcomesTotal no. of possible outcomes P\left( {{\text{Event}}} \right) = \dfrac{{{\text{No}}{\text{. of favourable outcomes}}}}{{{\text{Total no}}{\text{. of possible outcomes }}}}
The probability of drawing a red ball, P(R)=n(R)6+xP\left( R \right) = \dfrac{{n\left( R \right)}}{{6 + x}}
The probability of drawing a blue ball, P(B)=n(B)6+xP\left( B \right) = \dfrac{{n\left( B \right)}}{{6 + x}}
It is given in the problem that the probability of drawing a blue ball is twice that of a red ball, then according to the given statement:
P(B)=2×P(R)P\left( B \right) = 2 \times P\left( R \right)
Substitute the values of the probability:
n(B)6+x=2×n(R)6+x\dfrac{{n\left( B \right)}}{{6 + x}} = 2 \times \dfrac{{n\left( R \right)}}{{6 + x}}
Substitute the values n(B)=xn\left( B \right) = x and n(R)=6n\left( R \right) = 6 into the equation:
x6+x=2×66+x\dfrac{x}{{6 + x}} = 2 \times \dfrac{6}{{6 + x}}
Solve the equation for the value of xx.
x6+x=126+x\dfrac{x}{{6 + x}} = \dfrac{{12}}{{6 + x}}
⇒x=12\Rightarrow x = 12

It means that there are 12 blue balls in the bag.
Then the total number of balls in the bag are:
Total no. of balls, n(T)=6+x=6+12n\left( T \right) = 6 + x = 6 + 12

Total no. of balls, n(T)=18n\left( T \right) = 18

Thus, there are a total 1818 balls in the bag.

Note: The probability defines the numerical description about the happening of any event. It is equal to the ratio of the number of favorable outcomes to the total number of possible outcomes.