Calculating the Probability of Drawing 1 Red and 1 Black Ball from a Bag
Introduction
In this article, we will calculate the probability of drawing 1 red ball and 1 black ball from a bag containing 5 red balls and 4 black balls, without replacement of the balls. This calculation involves understanding the principles of combinatorial probability and the impact of drawing without replacement.
Step-by-Step Calculation
Total Balls
Ball Type Count Red 5 Black 4 Total 9Case 1: Picking a Red Ball First
1. Probability of picking a red ball first:
P(Red first) 5/9
2. After picking a red ball, there are now 8 balls left: 4 red and 4 black.
3. Probability of picking a black ball second:
P(Black second | Red first) 4/8 1/2
4. Combined probability for this case:
P(Red first then Black) P(Red first) × P(Black second | Red first) 5/9 × 1/2 5/18
Case 2: Picking a Black Ball First
1. Probability of picking a black ball first:
P(Black first) 4/9
2. After picking a black ball, there are now 8 balls left: 5 red and 3 black.
3. Probability of picking a red ball second:
P(Red second | Black first) 5/8
4. Combined probability for this case:
P(Black first then Red) P(Black first) × P(Red second | Black first) 4/9 × 5/8 20/72 5/18
Total Probability
Finally, we add the probabilities of both cases:
P(1 Red and 1 Black) P(Red first then Black) P(Black first then Red) 5/18 5/18 10/18 5/9
Conclusion
The probability of picking 1 red ball and 1 black ball in any order is:
boxed{5/9}
The Mathematics Behind the Calculation
The problem does not specify the order in which the balls are drawn, so it can be either “first draw Red ball then draw black ball” or “first draw Black ball then draw Red ball.”
The formula to calculate the probability is:
Pr[Red and Black] Pr[{first Red and then Black}] Pr[{first Black and then Red}] Pr[Red].Pr[Black | Red] Pr[Black].Pr[Red | Black] 4/9 × 5/8 5/9 × 4/8 2 × (4/9 × 5/8)
Pr[Red and Black] 5/9
Understanding Probability in Combinatorial Groupings
In scenarios where you need to draw items from a group without replacement, the order of the draws can affect the probability. This is why we calculated the probabilities for both possible orders and then summed them up.
The combinatorial approach used here ensures a comprehensive understanding of the problem by considering all possible sequences. This method is fundamental in probability theory and statistical analysis, making it highly relevant for SEO-content that aims to engage a technical and mathematical audience.