Probability of Drawing a White or Red Ball from a Box
The problem at hand involves calculating the probability of drawing either a white ball or a red ball from a box containing various colored balls. This scenario is a classic example of basic probability theory, which is an essential concept in both theoretical and applied mathematics.
Understanding the Problem
A box contains a total of 18 balls, consisting of 3 white balls, 6 red balls, and 9 blue balls. Our objective is to determine the probability of drawing a ball that is either white or red. To find this probability, we first need to understand the fundamental principles governing probabilities.
Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. In this case, the favorable outcomes are the events where a white or a red ball is drawn.
Calculating the Total Number of Balls
To begin, we calculate the total number of balls in the box:
Total number of balls 3 (white) 6 (red) 9 (blue) 18
Determining the Number of Favorable Outcomes
Next, we determine the number of favorable outcomes, which are the cases where we draw either a white ball or a red ball:
Number of favorable outcomes for white balls 3 Number of favorable outcomes for red balls 6The total number of favorable outcomes is the sum of the favorable outcomes for white and red balls:
Total favorable outcomes 3 (white) 6 (red) 9
Calculating the Probability
The probability of drawing a white ball or a red ball is therefore calculated as follows:
(P(text{White or Red}) frac{text{Number of favorable outcomes}}{text{Total number of outcomes}} frac{9}{18} frac{1}{2})
This simplifies to 0.5 or 50%. Thus, the probability of drawing either a white or a red ball from the box is 1/2 or 50%.
Verification and Application
This result can also be verified by considering the probabilities individually:
Probability of drawing a white ball ( frac{3}{18} frac{1}{6} ) Probability of drawing a red ball ( frac{6}{18} frac{2}{6} ) Total probability of drawing a white or red ball ( frac{1}{6} frac{2}{6} frac{3}{6} frac{1}{2} )Moreover, the combination approach using the binomial coefficient ( binom{9}{1} / binom{18}{1} ) also leads to the same result:
( frac{9}{18} frac{1}{2} )
Conclusion
In conclusion, the probability of drawing a white ball or a red ball from the box is ( frac{1}{2} ) or 50%. This result is consistent across different approaches and validates the initial calculation. Understanding such probability concepts is crucial in various fields, including statistical analysis, risk assessment, and decision-making processes.
Related Queries
Probability of drawing a white ball Probability of drawing a red ball Probability of drawing either white or red ballFor more in-depth coverage of probability theories and applications, explore related articles and tutorials available online.