Probability of Sequential Events in Candy Selection
Probability theory forms the backbone of understanding random events in our daily lives, such as selecting candies from a jar. Let's delve into the intriguing problem of finding the probability of choosing a dark chocolate, eating it, and then choosing a caramel. This process involves several steps and calculations, which we will cover in detail.
Problem Statement
A jar contains 2 caramels, 7 mints, and 16 dark chocolates. The question is: What is the probability of choosing a dark chocolate, eating it, and then choosing a caramel?
Step-by-Step Solution
To calculate the probability of these sequential events, we need to follow a step-by-step approach:
Step 1: Determine the Total Number of Candies
The total number of candies in the jar is calculated as follows:
Number of caramels: 2
Number of mints: 7
Number of dark chocolates: 16
Total 2 7 16 25 candies
Step 2: Calculate the Probability of Choosing a Dark Chocolate
The probability of choosing a dark chocolate is calculated by dividing the number of dark chocolates by the total number of candies:
Pdark chocolate (frac{16}{25})#160; 0.64 64%
Step 3: Determine the New Total Number of Candies
After eating one dark chocolate, the number of candies decreases by one, and the number of dark chocolates also decreases by one.
Total 25 - 1 24 candies
Step 4: Calculate the Probability of Choosing a Caramel After Eating the Dark Chocolate
The probability of choosing a caramel after eating the dark chocolate is calculated as follows:
Pcaramel | dark chocolate eaten (frac{2}{24})#160; (frac{1}{12})
Step 5: Calculate the Overall Probability
The overall probability of both events happening (choosing a dark chocolate, eating it, and then choosing a caramel) is the product of the two probabilities:
Pdark chocolate and then caramel Pdark chocolate × Pcaramel | dark chocolate eaten
Pdark chocolate and then caramel (frac{16}{25} times frac{1}{12} frac{16}{300} frac{4}{75})
The probability of choosing a dark chocolate, eating it, and then choosing a caramel is (frac{4}{75}).
Further Exploration
The concept of probability of sequential events is crucial in many real-world applications, including finance, quality control, and statistical analysis. Understanding how these probabilities are calculated can help in making informed decisions in various scenarios.
Similar Problem: Probability of Choosing Green Balls
As a reference point, consider a similar problem: A jar contains 6 red balls, 3 green balls, 5 white balls, and 7 yellow balls. If two balls are chosen from the jar with replacement, what is the probability that both balls chosen are green?
Step 1: Determine the Total Number of Balls
Total 6 3 5 7 21 balls
Step 2: Calculate the Probability of Choosing a Green Ball
Pgreen (frac{3}{21})#160; (frac{1}{7}) 14.29%
Step 3: Calculate the Probability of Choosing Two Green Balls with Replacement
Pgreen and green (with replacement) Pgreen × Pgreen
Pgreen and green (with replacement) (frac{1}{7} times frac{1}{7} frac{1}{49}) 2.04%
Conclusion
The probability theory applications in candy selection are not only educational but also practical. Understanding these concepts can provide insight into more complex real-world scenarios.