Problem 2 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

In a bag there are 5 white and 10 black balls. If ball is drawn at random from it, what is the probability that it is white?

Step-by-Step Solution

Answer

The probability of drawing a white ball is \( \frac{1}{3} \).

Step 1: Determine Total Number of Balls

To find the probability, we need to know the total number of balls in the bag. Add the number of white balls to the number of black balls: 5 white + 10 black = 15 total balls.

Step 2: Identify Favorable Outcomes

A favorable outcome for this probability question is drawing a white ball. Since there are 5 white balls, there are 5 favorable outcomes.

Step 3: Calculate Probability

Probability is calculated using the formula: \( \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). Substitute the numbers: \( \frac{5}{15} \).

Step 4: Simplify the Probability

To simplify \( \frac{5}{15} \), divide the numerator and the denominator by their greatest common divisor, which is 5: \( \frac{5 \div 5}{15 \div 5} = \frac{1}{3} \).

Key Concepts

Favorable Outcomes

When we talk about "favorable outcomes" in probability, we mean the specific results that align with the event we are interested in. In this exercise, the event is drawing a white ball from the bag. Hence, any instance where a white ball is drawn is considered a favorable outcome.

It is essential to clearly identify what counts as a favorable outcome before attempting to solve a probability problem. For example, if the problem asked for the likelihood of picking a black ball, then the number of black balls would constitute the number of favorable outcomes. However, for our specific problem, we care about white balls, so we count them, realizing there are 5 favorable outcomes.

These outcomes form the numerator in our probability calculation, serving as a key factor in determining how likely our event is, compared to all possible outcomes.

Total Number of Possible Outcomes

The total number of possible outcomes represents all the different results that can occur from an experiment or action. In our scenario, it is the total number of balls that could be drawn from the bag, regardless of their color.

Here, the total number of outcomes is simply the sum of white and black balls in the bag, which is 5 + 10 = 15. Remember that total possible outcomes include all events, regardless of whether they meet our criteria for success or not.

By identifying both the favorable and the total number of possible outcomes, we lay the foundation for calculating probability. The total number of outcomes fills the denominator in our probability equation.

Simplifying Fractions

After finding the fraction representing the probability, it's crucial to simplify it to its lowest terms to make the probability easier to interpret. Simplifying fractions involves reducing the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

In our problem, the fraction calculated is \( \frac{5}{15} \). Here, both 5 and 15 can be divided by 5, which is their GCD. So, \( \frac{5}{15} = \frac{5 \div 5}{15 \div 5} = \frac{1}{3} \). By doing this, we see that the probability of drawing a white ball is simplified to \( \frac{1}{3} \), indicating that about one-third of the time, a white ball will be drawn if done at random.

Simplifying fractions not only makes the number easier to understand and work with, but it also provides a clearer, more intuitive insight into the likelihood of the event occurring.