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Chapter 3: Problem 25

What is the probability of rolling one six-sided die and obtain ing the following numbers? a. 2 b. 1 or 2 c. An even number d. Any number but a 6

Short Answer

Expert verified
a. \( \frac{1}{6} \); b. \( \frac{1}{3} \); c. \( \frac{1}{2} \); d. \( \frac{5}{6} \)

Step by step solution

01

Understanding the Problem

We need to find the probability of specific outcomes when rolling a standard six-sided die. The die has six faces numbered from 1 to 6, and each outcome has an equal probability of occurring.
02

Finding Probability for a Specific Number (Part A)

To find the probability of rolling a 2, note that there is only one face with the number 2 on it. The probability is the number of favorable outcomes divided by the total possible outcomes. Therefore, the probability is \( \frac{1}{6} \).
03

Finding Probability for Multiple Numbers (Part B)

To find the probability of rolling a 1 or 2, we add the probabilities of each independent outcome: \( P(1) + P(2) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \).
04

Finding Probability for Even Numbers (Part C)

The even numbers on a die are 2, 4, and 6. Each appears on one face, so the number of favorable outcomes is 3. The probability is \( \frac{3}{6} = \frac{1}{2} \).
05

Finding Probability for Specific Condition (Part D)

Any number but a 6 includes 1, 2, 3, 4, and 5. There are 5 favorable outcomes. Thus, the probability of not rolling a 6 is \( \frac{5}{6} \).

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Key Concepts

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

Rolling a Six-Sided Die
A six-sided die, often used in board games, is a cube where each face is numbered from 1 to 6. When you roll a six-sided die, you could see any one of these numbers face up. Each of these faces has an equal chance of landing face up when the die is tossed. This means that every number from 1 to 6 has a probability of appearing of \( \frac{1}{6} \) every time you roll the die.
  • If you roll the die once, you are engaging in a single trial.
  • Each face of the die represents a possible outcome of this trial.
Understanding this basic setup helps in solving probability questions related to dice.
Favorable Outcomes
In probability, a favorable outcome is the outcome that you are interested in finding. When you roll a die, any specific number, like rolling a 2, can be considered a favorable outcome if that's what you're looking for. Suppose you want to find the probability of rolling a number greater than 3. Here, the favorable outcomes would be 4, 5, and 6. Suppose you seek the probability of rolling a 1 or 2; then these two numbers are your favorable outcomes.
  • The desired outcome, such as rolling a specific number, forms a subset of all possible outcomes.
  • The probability of a favorable outcome is calculated by dividing the number of favorable outcomes by the total number of possible outcomes, in this case, 6.
Independent Probability
Independent probability refers to a situation where the occurrence of one event does not affect the probability of another. When you roll a die multiple times, each roll is an independent event. The result of one roll does not change the outcome or probability of subsequent rolls. For example, if you rolled a 3 on your first roll, the chance of rolling a 2 on the next roll remains \( \frac{1}{6} \). Similarly, the probability of rolling 1, 2, or any other number remains the same for each independent roll.
  • Events are considered independent if the occurrence of one does not influence the other.
  • Each number has its own equal probability in independent scenarios.
Even Numbers on a Die
In the context of dice, even numbers are those divisible by 2 without leaving a remainder. On a six-sided die, the even numbers are 2, 4, and 6. If a problem asks for the probability of rolling an even number, the favorable outcomes are these three numbers.To calculate the probability of rolling an even number, count the even numbers on the die and divide by the total possible outcomes.\[\text{Probability (even number)} = \frac{\text{Number of even numbers}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2}\]
  • Even numbers offer a straightforward way to categorize outcomes on a die.
  • The probability simplifies since there are exactly half as many even numbers as total numbers on the die.

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Most popular questions from this chapter

In watermelons, bitter fruit \((B)\) is dominant over sweet fruit \((b),\) and yellow spots \((S)\) are dominant over no spots \((s) .\) The genes for these two characteristics assort independently. A homozygous plant that has bitter fruit and yellow spots is crossed with a homozygous plant that has sweet fruit and no spots. The \(F_{1}\) are intercrossed to produce the \(\mathrm{F}_{2}\) a. What are the phenotypic ratios in the \(\mathrm{F}_{2} ?\) b. If an \(F_{1}\) plant is backcrossed with the bitter, yellow-spotted parent, what phenotypes and proportions are expected in the offspring? c. If an \(F_{1}\) plant is backcrossed with the sweet, nonspotted parent, what phenotypes and proportions are expected in the offspring?

What characteristics of an organism would make it suitable for studies of the principles of inheritance? Can you name several organ isms that have these character istics?

A genet icist discovers an obese mouse in his laboratory colony, He breeds this ob ese mouse with a normal mouse. All the \(F_{1}\) mice from this cross are normal in size. When he interbreeds two \(\mathrm{F}_{1}\) mice, eight of the \(\mathrm{F}_{2}\) mice are normal in size and two are obese. The gencticist then intercrosses two of his obese mice, and he finds that all of the progeny from this cross are obese. These results lead the geneticist to conclude that obesity in mice results from a recessive allele. A second geneticist at a different university also discovers an obese mouse in her laboratory colony. She carries out the same crosses as those done by the first geneticist and obtains the same results. She also concludes that obesity in mice results from a recessive allele. One day the two geneticists meet at a genetics conference, learn of each other's experiments, and decide to exchange mice. They both find that, when they cross two obese mice from the different laboratories, all the offspring are normal; however, when they cross two obese mice from the same laboratory, all the offspring are obese. Explain their results.

White \((w)\) coat color in guinea pigs is recessive to black (W). In \(1909,\) W. E. Castle and J. C. Phillips transplanted. An ovary from a black guinea pig into a white female whose ovaries had been removed. They then mated this white female with a white male. All the offspring from the mating were black in color (W. E. Castle and J. C. Phillips. \(1909 .\) Science \(30: 312-313\) ). (IMAGES CANNOT COPY) a. Explain the results of this cross. b. Give the genotype of the offspring of this cross. c. What, if anything, does this experiment indicate about the validity of the pangenesis and the germ-plasm

In guinea pigs, the allele for black fur ( \(B\) ) is dominant over the allele for brown \((b)\) fur. A black guinea pig is crossed with a brown guinea pig, producing five \(\mathrm{F}_{1}\) black guinea pigs and six \(F_{1}\) brown guinea pigs. a. How many copies of the black allele \((B)\) will be present in each cell of an \(F_{1}\) black guinea pig at the following stages: \(G_{1}, G_{2},\) metaphase of mitosis, metaphase I of meiosis, metaphase II of meiosis, and after the second cytokinesis following meiosis? Assume that no crossing over takes place. b. How many copies of the brown allele \((b)\) will be present in each cell of an \(\mathrm{F}_{1}\) brown guinea pig at the same stages as those listed in part \(a\) ? Assume that no crossing over takes place.
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