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Chapter 4: Problem 21

Two fair dice are tossed. a. What is the probability that the sum of the number of dots shown on the upper faces is equal to \(7 ?\) To \(11 ?\) b. What is the probability that you roll "doubles" that is, both dice have the same number on the upper face? c. What is the probability that both dice show an odd number?

Short Answer

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Question: Find the probabilities of the following scenarios when two fair dice are tossed: a) the sum of the number of dots shown on the upper faces equal to 7 and 11, b) rolling doubles, c) both dice showing an odd number. Answer: The probabilities are a) \(P(7) = \frac{1}{6}\) and \(P(11) = \frac{1}{18}\), b) \(P(Doubles) = \frac{1}{6}\), and c) \(P(Odd\ numbers) = \frac{1}{4}\).

Step by step solution

01

Probability of Sum equal to 7 or 11

To solve this, we need to find all the possible combinations on the upper faces that would result in the sum being 7 or 11. For sum equal to 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) For sum equal to 11: (5, 6), (6, 5) Now, we know there are 36 possible outcomes when tossing two fair dice. The probability of an event occurring is the number of successful outcomes divided by the total number of possible outcomes. Probability of sum equal to 7: \(P(7) = \frac{6}{36} = \frac{1}{6}\) Probability of sum equal to 11: \(P(11) = \frac{2}{36} = \frac{1}{18}\)
02

Probability of Rolling Doubles

To find the probability of rolling doubles, we need to find all the possible combinations where both dice have the same number on the upper face. There are 6 possible combinations: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) The total number of possible outcomes is still 36. So, the probability of rolling doubles is: \(P(Doubles) = \frac{6}{36} = \frac{1}{6}\)
03

Probability of both dice showing odd numbers

Lastly, we need to find the probability of both dice showing an odd number. The possible combinations for this case are: (1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5) Again, there are 36 possible outcomes when tossing two fair dice. So the probability of both dice showing odd numbers is: \(P(Odd\ numbers) = \frac{9}{36} = \frac{1}{4}\) To summarize, the probabilities are: - Probability of the sum equal to 7: \(\frac{1}{6}\) - Probability of the sum equal to 11: \(\frac{1}{18}\) - Probability of rolling doubles: \(\frac{1}{6}\) - Probability of both showing odd numbers: \(\frac{1}{4}\)

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Key Concepts

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

Dice Probabilities
When tossing two fair dice, the outcome on each die is independent of the other. Each die has six faces, numbered from 1 to 6. This creates a total of 36 different combinations of outcomes when they are rolled together. Understanding the probabilities associated with dice rolls involves calculating the likelihood of various events.
  • To find the probability of a specific event, divide the number of favorable outcomes by the total number of possible outcomes.
  • An event, such as rolling a sum or doubles, is considered 'favorable' when its requirements are met with desired outcomes.
For example, finding the probability of the sum of the dice equals 7 or 11 requires counting the combinations that result in these sums, then dividing by 36.
Through this straightforward approach, we understand the chance of specific dice outcomes.
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations of objects. In this exercise, it helps to identify valid dice combinations that satisfy certain conditions, such as achieving specific sums. Each throw of two dice represents a pair in a sequence, with possible results like (2, 5) or (3, 4).
  • For sums like 7, distinct pairs are identified: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).
  • For a sum of 11, fewer pairs exist: (5, 6) and (6, 5).
  • To find doubles, look for identical pairs: (1, 1), (2, 2), etc.
Identifying these combinations by understanding the nature of dice and the possible arrangements they create is crucial for calculating probabilities accurately. Combinatorics provides the tools to efficiently enumerate all potential outcomes.
Events and Outcomes
When analyzing dice probabilities, it is vital to understand the concepts of events and outcomes.
An **event** is any specific occurrence we might be interested in; for example, the sum of the dice being 7 or rolling the same number on both dice.
  • An **outcome** is a potential result; when rolling two dice, (2, 3) or (5, 5) are examples of outcomes.
  • The set of all possible outcomes forms what is called the sample space鈥�36 different outcomes exist for two dice.
In our scenario, different events change the favorable outcomes set: - "Sum equals 7" event includes all pairs that add up to 7. - "Rolling doubles" event includes pairs where both dice show the same number. Clearly defining these concepts helps clarify what you are calculating and why. Each event requires a distinct set of outcomes for probability determination.
Basic Probability Concepts
Probability theory provides a mathematical framework to quantify uncertainty in various situations, such as rolling dice.
  • The **probability** of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
  • Probabilities are expressed as fractions between 0 and 1, where 0 means an event will not happen, and 1 means it will definitely happen.
For dice, understanding the nature of randomness and independence is key:- Each die is independent. The result of one does not affect the other.- The probability of multiple events occurring together is determined by their individual probabilities.In the problems given:- (For a sum of 7) out of 36 possible outcomes, only 6 result in 7, yielding a probability of \(\frac{1}{6}\).Analyses like these demonstrate how basic probability principles apply to practical scenarios, providing insights into seemingly random events.

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Most popular questions from this chapter

In \(1865,\) Gregor Mendel suggested a theory of inheritance based on the science of genetics. He identified heterozygous individuals for flower color that had two alleles \((\mathrm{r}=\) recessive white color allele and \(\mathrm{R}=\) dominant red color allele ). When these individuals were mated, \(3 / 4\) of the offspring were observed to have red flowers and \(1 / 4\) had white flowers. The table summarizes this mating; each parent gives one of its alleles to form the gene of the offspring. We assume that each parent is equally likely to give either of the two alleles and that, if either one or two of the alleles in a pair is dominant (R), the offspring will have red flowers. a. What is the probability that an offspring in this mating has at least one dominant allele? b. What is the probability that an offspring has at least one recessive allele? c. What is the probability that an offspring has one recessive allele, given that the offspring has red flowers?

A student prepares for an exam by studying a list of 10 problems. She can solve 6 of them. For the exam, the instructor selects 5 questions at random from the list of \(10 .\) What is the probability that the student can solve all 5 problems on the exam?

A woman brought a complaint of gender discrimination to an eight-member HR committee. The committee, composed of five females and three males, voted \(5-3\) in favor of the woman, the five females voting for the woman and the three males against. Has the board been affected by gender bias? That is, if the vote in favor of the woman was \(5-3\) and the board members were not biased by gender, what is the probability that the vote would split along gender lines (five females for, three males against)?

Suppose \(P(A)=.1\) and \(P(B)=.5 .\) $$\text { If } P(A \cup B)=.65, \text { are } A \text { and } B \text { mutually }$$$$\text { exclusive? }$$

In a genetics experiment, the researcher mated two Drosophila fruit flies and observed the traits of 300 offspring. The results are shown in the table. $$ \begin{array}{lcc} \hline & \quad\quad\quad {\text { Wing Size }} \\ { 2 - 3 } \text { Eye Color } & \text { Normal } & \text { Miniature } \\ \hline \text { Normal } & 140 & 6 \\ \text { Vermillion } & 3 & 151 \end{array} $$ One of these offspring is randomly selected and observed for the two genetic traits. a. What is the probability that the fly has normal eye color and normal wing size? b. What is the probability that the fly has vermillion eyes? c. What is the probability that the fly has either vermillion eyes or miniature wings, or both?
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