Probability of Selecting Red or Yellow Marbles from a Box

Introduction

Middle school students often face probability problems, which involve understanding the likelihood of specific events occurring. One common type of problem involves determining the probability of selecting a particular colored marble from a box. In this article, we will explore various scenarios where a box contains different colored marbles and calculate the probability of selecting a red or yellow marble.

Scenario 1: 44 Marbles in a Box

Consider a box containing a total of 44 marbles, out of which 14 are red, 12 are blue, and 18 are yellow. Let's explore the probability of picking a red or yellow marble at random.

Step-by-Step Calculation

Number of red or yellow marbles Number of red marbles Number of yellow marbles

14 18 32

[ text{Probability} frac{text{Number of favorable outcomes}}{text{Total number of outcomes}} frac{32}{44} ]

Alternatively, this can be simplified as:

[ frac{32}{44} frac{8}{11} ]

Therefore, the probability that Dario selects a red marble or a yellow marble is:

[ boxed{frac{8}{11}} ]

Scenario 2: Different Total Marbles

Let's consider a more complex scenario where an urn contains 12 red marbles, 20 blue marbles, and 42 yellow marbles. The total number of marbles in the urn is 74.

Step-by-Step Calculation

The total number of favorable outcomes (red or blue marbles) is:

[ 12 20 32 ]

Thus, the probability of picking a red or blue marble is:

[ text{Probability} frac{32}{74} frac{16}{37} approx 0.4324 ]

This probability can also be calculated using the complement rule:

[ text{Probability of not picking yellow} 1 - frac{42}{74} frac{32}{74} ]

Conclusion

Understanding and calculating probabilities is fundamental in probability theory. In this article, we analyzed two scenarios involving different numbers of marbles and provided step-by-step calculations to determine the probability of selecting a red or yellow marble. These exercises are essential for developing a strong foundation in probability, a subject widely applicable in fields such as mathematics, statistics, and data analysis.

Further Reading

Interested readers can explore further resources on probability, such as the complement rule, mutually exclusive events, and more complex probability problems. Such knowledge enhances critical thinking and problem-solving skills.