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Chapter 5: Problem 77

Box of chocolates According to Forrest Gump, Life is like a box of chocolates. You never know what you鈥檙e gonna get. Suppose a candy maker offers a special Gump box with 20 chocolate candies that look the same. In fact, 14 of the candies have soft centers and 6 have hard centers. Choose 2 of the candies from a Gump box at random. (a) Draw a tree diagram that shows the sample space of this chance process. (b) Find the probability that one of the chocolates has a soft center and the other one doesn鈥檛.

Short Answer

Expert verified
The probability that one chocolate is soft and the other is hard is \( \frac{42}{95} \).

Step by step solution

01

Tree Diagram Setup

Create the initial state for the tree diagram. Here, the first level of the tree will represent the selection of the first candy. You have two branches: 'Soft Center' (14 possibilities out of 20) and 'Hard Center' (6 possibilities out of 20).
02

Second Selection Branches

For each first candy choice, create branches for choosing the second candy. If the first candy was 'Soft', there are 13 'Soft' and 6 'Hard' candies left. If the first candy was 'Hard', there are still 14 'Soft' and 5 'Hard' candies left. This forms the second level of the tree.
03

Calculate Probabilities for Soft then Hard

Calculate the probability of selecting a 'Soft' candy followed by a 'Hard' candy. Start with the initial probability of selecting a 'Soft' candy: \( \frac{14}{20} \). Then multiply by the probability of selecting a 'Hard' candy from the remaining: \( \frac{6}{19} \). The probability is \( \frac{14}{20} \times \frac{6}{19} = \frac{84}{380} \).
04

Calculate Probabilities for Hard then Soft

Calculate the probability of selecting a 'Hard' candy followed by a 'Soft' candy. Begin with the initial probability of selecting a 'Hard' candy: \( \frac{6}{20} \). Then multiply by the probability of selecting a 'Soft' candy from the remaining: \( \frac{14}{19} \). The probability is \( \frac{6}{20} \times \frac{14}{19} = \frac{84}{380} \).
05

Combine Probabilities

Add the probabilities of the favorable outcomes: selecting one 'Soft' and one 'Hard' candy. This is \( \frac{84}{380} + \frac{84}{380} = \frac{168}{380} \).
06

Simplify the Probability

Reduce the fraction \( \frac{168}{380} \) to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 4. Thus, \( \frac{168}{380} = \frac{42}{95} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tree Diagram
A tree diagram is a visual representation that helps us outline all the possible outcomes of a chance process. It's particularly useful in problems related to probability and decision-making. In the example of selecting chocolates from a Gump box, we visualize each step of the selection with branches.

For our problem, the first level of our tree diagram depicts the choice of a chocolate candy: a 'Soft Center' or a 'Hard Center'. Each "branch" shows these options along with their probabilities, picturing the initial state of the selection.
  • First Branch: Soft Center (14 possibilities out of 20)
  • Second Branch: Hard Center (6 possibilities out of 20)
The second level features the selection of another candy, adjusting to the candy already chosen. This visualization allows a structured examination of all possible scenarios of picking the candies in sequence.
Sample Space
The sample space in probability is the set of all possible outcomes. It's crucial because it lays the groundwork for calculating probabilities. In the Gump box example, the sample space includes all potential combinations of two candy selections.

Consider a step-by-step breakdown via the tree diagram. Each path from the tree's root to its leaves corresponds to an outcome in the sample space. For example, one outcome is picking a 'Soft' candy first and then a 'Hard' candy, and vice versa. This tree-based approach ensures we don't miss out on any potential combinations, providing a comprehensive view of all possible outcomes.
  • Soft then Hard
  • Hard then Soft
  • Soft then Soft
  • Hard then Hard
However, for our probability calculation, our focus is on paths where one candy is soft and the other is hard.
Step-by-Step Probability
Step-by-step probability is a methodical approach to finding the likelihood of certain outcomes by breaking down problems into smaller, manageable steps. This method is handy to ensure clarity and prevent mistakes.

In our problem, we first compute the probability of each individual path from the tree. For instance, the probability of the first candy being 'Soft' followed by a 'Hard' candy is calculated by multiplying their respective probabilities:
\[\frac{14}{20} \times \frac{6}{19} = \frac{84}{380}\]
Similarly, for selecting a 'Hard' candy first and then a 'Soft' candy:
\[\frac{6}{20} \times \frac{14}{19} = \frac{84}{380}\]

Adding these probabilities gives the total probability of picking two candies where one is 'Soft' and one is 'Hard':
\[\frac{84}{380} + \frac{84}{380} = \frac{168}{380}\]
This breakdown into steps makes challenging tasks more approachable, ensuring no detail is overlooked.
Combinatorics
Combinatorics is a branch of mathematics that focuses on counting, arranging, and combining objects in specified manners. It plays an integral role in understanding probability problems like the one with chocolates.

Our challenge involves selecting two candies from a box of 20, where the choice is between 'Soft' and 'Hard' centers. Combinatorics helps us understand how to compute various outcomes without duplication or omission.
  • We use combinatorial reasoning to determine the different ways to select one 'Soft' and one 'Hard' candy while acknowledging that order matters (Soft then Hard, Hard then Soft).
  • This involves recognizing not all combinations of the two choices are favorable (Soft-Soft and Hard-Hard do not meet the criteria).
Combinatorial calculations aid in laying a solid foundation for understanding the structure of probability spaces and ensure precise calculations.

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Most popular questions from this chapter

Mac or PC? A recent census at a major university revealed that 40% of its students mainly used Macintosh computers (Macs). The rest mainly used PCs. At the time of the census, 67% of the school鈥檚 students were undergraduates. The rest were graduate students. In the census, 23% of the respondents were graduate students who said that they used PCs as their primary computers. Suppose we select a student at random from among those who were part of the census and learn that the student mainly uses a Mac. Find the probability that this person is a graduate student. Show your work.

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Multiple choice: Select the best answer for Exercises 104 to 106. An athlete suspected of using steroids is given two tests that operate independently of each other. Test A has probability 0.9 of being positive if steroids have been used. Test B has probability 0.8 of being positive if steroids have been used. What is the probability that neither test is positive if steroids have been used? $$ \begin{array}{ll}{\text { (a) } 0.72} & {\text { (c) } 0.02 \quad \text { (e) } 0.08} \\ {\text { (b) } 0.38} & {\text { (d) } 0.28}\end{array} $$

Urban voters The voters in a large city are 40% white, 40% black, and 20% Hispanic. (Hispanics may be of any race in official statistics, but here we are speaking of political blocks.) A mayoral candidate anticipates attracting 30% of the white vote, 90% of the black vote, and 50% of the Hispanic vote. Draw a tree diagram to represent this situation. What percent of the overall vote does the candidate expect to get? Use the four-step process to guide your work.

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