Understanding Probability in the Context of Marble Selection

When it comes to probability, especially in the context of selecting items from a collection, replacement can significantly affect the outcome. This article delves into the probability of selecting specific marbles from a bag, both with and without replacement. We will dissect the problem step-by-step and understand the underlying principles of probability and independent events.

Introduction to the Problem

Consider a bag containing various marbles of different colors: 5 black, 7 blue, 3 green, 4 red, and 1 white. The question asks, what is the probability of selecting a black marble and then a red marble, considering the marble is replaced after the first selection. Additionally, we will explore the scenario where the first marble is not replaced.

Calculating Probability with Replacement

Step 1: Determine the Total Number of Marbles

First, we calculate the total number of marbles in the bag:

5 (black) 7 (blue) 3 (green) 4 (red) 1 (white) 20 marbles

Step 2: Calculate the Probability of Selecting a Black Marble

The probability of selecting a black marble on the first draw is calculated as follows:

[ P(text{Black}) frac{text{Number of Black Marbles}}{text{Total Number of Marbles}} frac{5}{20} frac{1}{4} ]

Step 3: Calculate the Probability of Selecting a Red Marble

Since the marble is replaced, the total number of marbles remains 20. The probability of selecting a red marble on the second draw is:

[ P(text{Red}) frac{text{Number of Red Marbles}}{text{Total Number of Marbles}} frac{4}{20} frac{1}{5} ]

Step 4: Calculate the Combined Probability

Since the selections are independent due to replacement, we multiply the probabilities:

[ P(text{Black then Red}) P(text{Black}) times P(text{Red}) frac{1}{4} times frac{1}{5} frac{1}{20} ]

Therefore, the probability of selecting a black marble and then a red marble, with replacement, is ( frac{1}{20} ).

Calculating Probability without Replacement

Step 1: Initial Probability of Selecting a Black Marble

Initially, the probability of selecting a black marble is:

[ P_1(text{Black}) frac{5}{20} frac{1}{4} ]

Step 2: Adjusted Probability of Selecting a Red Marble

Since the first marble is not replaced, the total number of marbles is now 19. The probability of selecting a red marble on the second draw is:

[ P_2(text{Red}) frac{4}{19} ]

Step 3: Calculate the Combined Probability without Replacement

The combined probability of both events occurring in sequence without replacement is the product of the two probabilities:

[ P(text{Black then Red}) P_1(text{Black}) times P_2(text{Red}) frac{1}{4} times frac{4}{19} frac{1}{19} ]

Therefore, the probability of selecting a black marble and then a red marble, without replacement, is ( frac{1}{19} ).

Conclusion

This problem illustrates the importance of understanding the conditions under which probability calculations are made. Whether with or without replacement, the principles of independent events must be considered. Remember, the key to solving probability problems lies in carefully analyzing the given conditions and applying the appropriate formulas.