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Chapter 5: Problem 75

Box of chocolates According to Forrest Gump, "Life is like a box of chocolates. You never know what you're 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't.

Short Answer

Expert verified
The probability of selecting one soft and one hard chocolate is \( \frac{42}{95} \).

Step by step solution

01

Understand the Sample Space

The Gump box contains 20 chocolates: 14 with soft centers and 6 with hard centers. When selecting 2 candies from the box, various combinations are possible. We need to explore these using a tree diagram to understand all potential outcomes.
02

Drawing the Tree Diagram

Begin the tree diagram with a first choice leading to two branches: one for choosing a soft and the other for a hard chocolate. From each of these branches, add two more branches representing a second selection. This results in four endpoint branches corresponding to four outcomes: soft-soft, soft-hard, hard-soft, and hard-hard.
03

Determining Outcome Probabilities

The probability of each branch or outcome can be found by calculating the probabilities as follows: - Probability of soft-soft: \( \frac{14}{20} \cdot \frac{13}{19} = \frac{182}{380} \)- Probability of soft-hard: \( \frac{14}{20} \cdot \frac{6}{19} = \frac{84}{380} \)- Probability of hard-soft: \( \frac{6}{20} \cdot \frac{14}{19} = \frac{84}{380} \)- Probability of hard-hard: \( \frac{6}{20} \cdot \frac{5}{19} = \frac{30}{380} \).Ensure the sum of probabilities equals 1 as a sanity check.
04

Calculating Required Probability

To find the probability of selecting one soft and one hard chocolate, combined probabilities of the soft-hard and hard-soft outcomes since order does not matter: \( \frac{84}{380} + \frac{84}{380} = \frac{168}{380} \). Simplifying this fraction gives \( \frac{84}{190} \) or further to \( \frac{42}{95} \).

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Key Concepts

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

Sample Space
The sample space in probability is the set of all possible outcomes. For our box of chocolates, the sample space reflects the different combinations of candies we can select. We have 20 chocolates with 14 soft and 6 hard centers. When we choose two chocolates, every pair represents an item in our sample space.

The combinations for two chocolates from the Gump box include:
  • Soft-Soft: Both chocolates have soft centers.
  • Soft-Hard: One chocolate has a soft center, the other a hard center.
  • Hard-Soft: First chocolate has a hard center, second one soft.
  • Hard-Hard: Both chocolates have hard centers.
By listing all these possibilities, we have effectively created our sample space. In probability problems, understanding the sample space is crucial as it forms the foundation for calculating probabilities.
Tree Diagram
A tree diagram is a visual representation of all possible outcomes in a probability scenario. It helps organize complex probability problems by breaking them down into a series of sequential random choices.

For the Gump box exercise:
  • Start with a single point, which represents the box before any candy is selected.
  • The first level of branches represents the two choices for the first candy: soft or hard.
  • Each branch then splits again for the second choice, showcasing all possible sequences of choices.
The resulting diagram aids in visualizing all scenarios like soft-soft, soft-hard, hard-soft, and hard-hard. Each path through the tree represents a distinct outcome, making complex selections simpler to manage.
Combinatorial Probability
Combinatorial probability involves calculating the likelihood of various outcomes by counting combinations and arrangements of events. For our chocolates, we calculate the probability of each event based on combinations of soft and hard-centered candies.

Each probability is derived by multiplying the fraction of choosing each candy at each step of the selection. For example:
  • Soft-Soft: First is soft, and second is soft, \[\frac{14}{20} \times \frac{13}{19} = \frac{182}{380}.\]
  • Soft-Hard: First is soft, and second is hard, \[\frac{14}{20} \times \frac{6}{19} = \frac{84}{380}.\]
  • Hard-Soft: First is hard, and second is soft, \[\frac{6}{20} \times \frac{14}{19} = \frac{84}{380}.\]
  • Hard-Hard: First is hard, and second is hard, \[\frac{6}{20} \times \frac{5}{19} = \frac{30}{380}.\]
This approach lets us see the proportion of each event relative to all possible selections.
Conditional Probability
Conditional probability is about finding the probability of an event given that another event has already occurred. In the chocolate box problem, after choosing the first candy, the available combinations for the second choice adjust based on this first selection.

Consider calculating probabilities for mixed selection:
  • The probability of one soft and one hard chocolate is the sum of the soft-hard and hard-soft paths in the tree: \[\frac{84}{380} + \frac{84}{380} = \frac{168}{380}.\]
  • This fraction simplifies to \[\frac{42}{95}.\]
Conditional probability helps answer questions about these dependent outcomes by focusing on what's left in the sample space after one event occurs.

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Most popular questions from this chapter

Languages in Canada Canada has two official languages, English and French. Choose a Canadian at random and ask, "What is your mother tongue?" Here is the distribution of responses, combining many separate languages from the broad Asia/Pacific region: $$\begin{array}{lcccc}\hline \text { Language: } & \text { English } & \text { French } & \text { Asian/Pacific } & \text { 0ther } \\\\\text { Probability: } & 0.63 & 0.22 & 0.06 & ? \\\\\hline\end{array}$$ (a) What probability should replace "?" in the distribution? Why? (b) What is the probability that a Canadian's mother tongue is not English? (c) What is the probability that a Canadian's mother tongue is a language other than English or French?

The probability of a flush A poker player holds a flush when all 5 cards in the hand belong to the same suit. We will find the probability of a flush when 5 cards are dealt. Remember that a deck contains 52 cards, 13 of each suit, and that when the deck is well shuffled, each card dealt is equally likely to be any of those that remain in the deck. (a) We will concentrate on spades. What is the probability that the first card dealt is a spade? What is the conditional probability that the second card is a spade given that the first is a spade? (b) Continue to count the remaining cards to find the conditional probabilities of a spade on the third, the fourth, and the fifth card given in each case that all previous cards are spades. (c) The probability of being dealt 5 spades is the product of the five probabilities you have found. Why? What is this probability? (d) The probability of being dealt 5 hearts or 5 diamonds or 5 clubs is the same as the probability of being dealt 5 spades. What is the probability of being dealt a flush?

Is this valid? Determine whether each of the following simulation designs is valid. Justify your answer. (a) According to a recent survey, \(50 \%\) of people aged 13 and older in the United States are addicted to texting. To simulate choosing a random sample of 20 people in this population and seeing how many of them are addicted to texting, use a deck of cards. Shuffle the deck well, and then draw one card at a time. A red card means that person is addicted to texting; a black card means he isn't. Continue until you have drawn 20 cards (without replacement) for the sample. (b) A tennis player gets \(95 \%\) of his serves in play during practice (that is, the ball doesn't go out of bounds). To simulate the player hitting 5 serves, look at 5 pairs of digits going across a row in Table \(D .\) If the number is between 00 and 94 , the serve is in; numbers between 95 and 99 indicate that the serve is out.

Late shows Some TV shows begin after their scheduled times when earlier programs run late. According to a network's records, about \(3 \%\) of its shows start late. To find the probability that three consecutive shows on this network start on time, can we multiply (0.97)(0.97)(0.97)\(?\) Why or why not?

Rolling dice Suppose you roll two fair, six-sided dice-one red and one green. Are the events "sum is \(7 "\) and "green die shows a 4 " independent? Justify your answer.
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