Understanding the Probability of Drawing a Black Ball

In a common probability problem, you are often asked to determine the likelihood of an event occurring. One such scenario involves drawing a ball from a bag containing a specific variety of balls. Here, we will explore how to calculate the probability of drawing a black ball from a bag containing a certain number of black and white balls. This guide will cover the steps required to derive the probability and will also include real examples for clarity.

Example 1: A Bag with 9 Black and 12 White Balls

Consider a bag that contains 9 black balls and 12 white balls. We want to find the probability of drawing a black ball at random. The probability formula is as follows:

[ P(text{black ball}) frac{text{Number of black balls}}{text{Total number of balls}} ]

In this case, the number of black balls is 9, and the total number of balls is 9 12, which is 21. Hence, the probability of drawing a black ball can be calculated as follows:

[ P(text{black ball}) frac{9}{21} ]

This fraction can be simplified:

[ frac{9}{21} frac{3}{7} ]

Therefore, the probability that the ball drawn is black is (frac{3}{7}).

Example 2: Alternative Method of Probability Calculation

Let's solve the same problem using another approach. The bag contains a total of 9 black and 12 white balls, amounting to 21 balls. One ball can be drawn from the bag in 21 ways. We want to find the probability that the drawn ball is black. There are 9 black balls, and one black ball can be chosen in 9 ways. Hence, the required probability is:

[ text{Required probability} frac{9}{21} frac{3}{7} ]

Example 3: Drawing a Second Black Ball from a Bag

Consider a bag that contains 12 black and 6 white balls, totaling 18 balls. If a black ball is drawn first, then there are 17 balls left, with 11 of them being black. The probability that the second ball drawn is black can be calculated as follows:

[ text{Probability} frac{11}{17} ]

Therefore, the probability that the second ball drawn is black is (frac{11}{17}).

Example 4: Another Scenarios Involving Different Numbers of Balls

In another example, the bag contains 64 black and 8 white balls, making a total of 72 balls. If the goal is to find the probability of drawing a white ball, the calculation would be:

[ text{Probability of white} frac{8}{72} frac{1}{9} ]

This simplifies to (frac{1}{9}), indicating that the probability of drawing a white ball is (frac{1}{9}).

Example 5: Drawing a Black Ball from a Bag Containing Different Numbers of Balls

Another example involves a bag containing 6 black and 8 white balls, making a total of 14 balls. If a black ball is drawn, the probability that the second ball drawn is also black is:

[ text{Probability} frac{6}{14} frac{3}{7} ]

This simplifies to (frac{3}{7}), showing that there is a 6 out of 14 chance, or 3 out of 7 chance, of drawing a black ball again.

Conclusion

In summary, the probability of drawing a black ball from a bag containing a certain number of black and white balls can be calculated using the formula (frac{text{Number of black balls}}{text{Total number of balls}}). This formula can be applied to a wide range of scenarios to determine the probability of drawing a black ball. Whether you are working with 9 and 12, 12 and 6, 64 and 8, or any other combination, the method remains consistent and straightforward.

Understanding and applying this concept can be useful in various fields, including statistics, probability theory, and even in practical situations where random sampling is involved.