Question Video: Using Theoretical Probability to Solve Problems | Nagwa Question Video: Using Theoretical Probability to Solve Problems | Nagwa

# Question Video: Using Theoretical Probability to Solve Problems Mathematics • Second Year of Preparatory School

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A bag contains 24 white balls and an unknown number of red balls. The probability of choosing at random a red ball is 7/31. How many balls are there in the bag?

03:11

### Video Transcript

A bag contains 24 white balls and an unknown number of red balls. The probability of choosing at random a red ball is seven over 31. How many balls are there in the bag?

We are told that this bag contains 24 white balls and an unknown number of red balls. We are also told that the probability of choosing a red ball from the bag is seven over 31.

We can recall that, in general, the probability of a particular event occurring can be found by dividing the number of successful outcomes by the total number of outcomes. This means that the probability of choosing a red ball from the bag would’ve been calculated by dividing the number of red balls in the bag by the total number of balls in the bag. We don’t know either of these values though. So we need to introduce some algebra to help us solve the problem.

We know the bag contains 24 white balls. We don’t know the number of red balls, so we can use the letter 𝑛 to represent this. The total number of balls in the bag can then be represented by the expression 𝑛 plus 24. We can then form an equation by substituting the given probability of seven over 31, 𝑛 for the number of red balls and 𝑛 plus 24 for the total number of balls, giving seven over 31 equals 𝑛 over 𝑛 plus 24.

To solve this equation for 𝑛, we begin by cross multiplying, giving seven multiplied by 𝑛 plus 24 is equal to 31𝑛. Distributing the parentheses on the left-hand side gives seven 𝑛 plus 168 equals 31𝑛. And then we can collect the like terms on the right-hand side of the equation by subtracting seven 𝑛 from each side to give 168 equals 24𝑛. Finally, we can divide both sides of the equation by 24 to give 𝑛 equals 168 over 24, which is seven. We now know that there are seven red balls in the bag, and so there are seven plus 24 balls in total. There are therefore 31 balls in the bag.

Now, if you’re wondering why we couldn’t have just equated the numerators and denominators of the two fractions, which would have given the same answer, it’s important to realize that whilst this would have worked for this specific set of numbers, it won’t work in every case. If the fraction for the probability had been simplified from its initial form, then equating the numerators and denominators would give inconsistent equations that we wouldn’t be able to solve for 𝑛. Following the formal method of solving the equation ensures we obtain the correct answer, which is 31.

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