Probability of Picking Blue Marbles with Replacement

In this article, we explore the concept of probability through an example that involves picking marbles from a bag with replacement. We will consider a scenario where a bag contains 7 red marbles, 6 blue marbles, 5 white marbles, and 2 green marbles. Our objective is to determine the probability of picking three blue marbles consecutively with replacement. Let's break down the steps and provide a detailed solution.

Scenario Overview

The bag contains a total of 20 marbles, with 6 of them being blue. The task is to find the probability of picking a blue marble three times in a row, with each marble being replaced after it is picked.

Understanding Replacement

Replacement in probability means that after a marble is selected, it is put back into the bag before the next pick. This ensures that the total number of marbles and their distribution remain the same for each pick.

Probability of a Single Blue Marble Pick

The probability of picking a blue marble in a single attempt is given by the ratio of the number of blue marbles to the total number of marbles in the bag.

Number of blue marbles: 6

Total number of marbles: 20

Probability (P) of picking a blue marble: [ P frac{6}{20} frac{3}{10} ]

Probability of Three Blue Marbles with Replacement

Since the marble is replaced each time, the probability remains the same for all three picks. To find the probability of picking three blue marbles consecutively, we multiply the probabilities of each individual event.

Probability of first blue pick: (frac{3}{10})

Probability of second blue pick: (frac{3}{10})

Probability of third blue pick: (frac{3}{10})

Probability of three blue marbles with replacement:

[ P left(frac{3}{10}right) times left(frac{3}{10}right) times left(frac{3}{10}right) left(frac{3}{10}right)^3 ]

Calculating the resulting probability: [ left(frac{3}{10}right)^3 frac{27}{1000} ]

This simplifies to:

[ frac{27}{1000} 0.027 quad text{or} quad 2.7%]

Conclusion

The probability of picking three blue marbles with replacement from the given bag is 0.027, or 2.7%. This calculation demonstrates the concept of probability with replacement, where each event is independent and the outcomes are calculated by multiplying the individual probabilities.

Key Takeaways

Understanding the concept of replacement in probability is crucial. The probability of multiple events with replacement is found by multiplying the probabilities of individual events. Replacement ensures that the total number of marbles and their distribution remain constant for each pick.

By applying these principles, one can solve similar probability problems involving replacement effectively.