Solving Probability Problems with Marbles: A Detailed Guide

Understanding how to calculate probabilities is a crucial skill in data analysis, statistics, and other mathematical fields. This article will walk you through a specific problem involving marbles, illustrating the key steps in solving probability problems. We will break down the problem and show how to calculate the total number of marbles in a bag given certain probabilities.

Given Probabilities and Information

The problem involves a bag of marbles that includes red, blue, and yellow marbles. The key information provided is:

The probability of picking a red marble is 7/15. The probability of picking a yellow marble is 20% or 1/5. There are 15 blue marbles in the bag.

Setting Up the Problem

We can use these probabilities to set up equations that will help us determine the total number of marbles in the bag. Let's define the following variables:

nR number of red marbles nB number of blue marbles 15 nY number of yellow marbles

The total number of marbles is R B Y nR nB nY.

Step-by-Step Solution

Step 1: Setting Up the Equations

From the probability of picking a red marble:

[ frac{n_{R}}{n_{R} n_{B} n_{Y}} frac{7}{15} ]

Cross-multiplying gives:

[ 15n_{R} 7n_{R} 7n_{B} 7n_{Y} ]

Since nB 15, the equation becomes:

[ 15n_{R} 7n_{R} 105 7n_{Y} ]

Expanding and simplifying:

[ 15n_{R} 7n_{R} 105 7n_{Y} ] [ 8n_{R} - 7n_{Y} 105 quad text{Equation 1} ]

From the probability of picking a yellow marble:

[ frac{n_{Y}}{n_{R} n_{B} n_{Y}} frac{1}{5} ]

Cross-multiplying gives:

[ 5n_{Y} n_{R} 15 n_{Y} ]

Rearranging terms:

[ 4n_{Y} n_{R} - 15 quad text{Equation 2} ]

Step 2: Solving the Equations

From Equation 2, we can express nR in terms of nY:

[ n_{R} 4n_{Y} 15 ]

Substitute nR into Equation 1:

[ 8(4n_{Y} 15) - 7n_{Y} 105 ]

Expanding and simplifying:

[ 32n_{Y} 120 - 7n_{Y} 105 ]

[ 25n_{Y} 225 ]

Dividing by 25:

[ n_{Y} 9 ]

Substitute nY 9 back into nR from Equation 2:

[ n_{R} 4(9) 15 36 15 21 ]

Step 3: Calculating the Total Number of Marbles

The total number of marbles is:

[ n_{R} n_{B} n_{Y} 21 15 9 45 ]

Conclusion

The total number of marbles in the bag is 45.

Additional Insights

The probability of picking a yellow marble is 1/5, which can be converted to a decimal (0.2) or a fraction (1/5) for easier calculations. The probability of picking a red marble is 7/15. Given that the total number of marbles is 45, the probability of picking a blue marble can be calculated as:

[ 1 - left( frac{7}{15} frac{3}{15} right) 1 - frac{10}{15} frac{5}{15} frac{1}{3} ]

This confirms that 1/3 of the marbles are blue, which is consistent with the given information.

To summarize, the key steps in solving the problem are:

Define the variables and set up the equations. Solve the equations to find the number of each type of marble. Calculate the total number of marbles.

This approach can be applied to similar probability problems involving different colors or types of objects.

Keywords: marble probability, probability calculation, math problems