A box contains white, milk, and dark chocolates. The ratio of white to milk chocolates is [tex]$2: 3$[/tex]
A chocolate is picked at random. The probability that it is a white chocolate is [tex]$\frac{2}{11}$[/tex]
What is the probability that it is a dark chocolate? Give your answer as a fraction in its simplest form.
Answer (2)
Let the number of white chocolates be 2 x , milk chocolates be 3 x , and dark chocolates be y .
The probability of picking a white chocolate is 5 x + y 2 x = 11 2 .
Solving for y gives y = 6 x .
The probability of picking a dark chocolate is 11 x 6 x = 11 6 .
Explanation
Understand the problem and provided data We are given that the ratio of white to milk chocolates is 2 : 3 . This means that for every 2 white chocolates, there are 3 milk chocolates. We are also given that the probability of picking a white chocolate is 11 2 . Our goal is to find the probability of picking a dark chocolate.
Define variables Let the number of white chocolates be 2 x and the number of milk chocolates be 3 x . Let the number of dark chocolates be y . The total number of chocolates is 2 x + 3 x + y = 5 x + y .
Express probability of picking white chocolate The probability of picking a white chocolate is the number of white chocolates divided by the total number of chocolates, which is 5 x + y 2 x = 11 2 .
Solve for y in terms of x Now we solve for y in terms of x . Cross-multiplying, we get 22 x = 10 x + 2 y . Subtracting 10 x from both sides gives 12 x = 2 y . Dividing both sides by 2 gives y = 6 x .
Calculate total number of chocolates The total number of chocolates is 5 x + y = 5 x + 6 x = 11 x .
Calculate probability of picking dark chocolate The probability of picking a dark chocolate is the number of dark chocolates divided by the total number of chocolates, which is 11 x y = 11 x 6 x = 11 6 .
State the final answer Therefore, the probability of picking a dark chocolate is 11 6 .
Examples
This type of probability problem can be used in real-life scenarios such as quality control in a chocolate factory. For example, if a factory produces chocolates with different ratios of white, milk, and dark chocolates, they can use this method to determine the probability of picking a specific type of chocolate. This can help them ensure that the chocolates are mixed in the correct proportions and that the quality of the product is consistent. Also, understanding ratios and probabilities is crucial in various fields like finance, statistics, and data analysis, where proportions and likelihoods are frequently analyzed to make informed decisions.
The probability of picking a dark chocolate is 11 6 . This is calculated by establishing the ratios of different chocolates and using the provided probability of picking a white chocolate. We define variables, set up equations, and solve for the probability of dark chocolates based on the known quantities.
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