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Chapter 4: Problem 1

Notation When randomly selecting an adult, \(A\) denotes the event of selecting someone with blue eyes. What do \(P(A)\) and \(P(\bar{A})\) represent?

Short Answer

Expert verified
P(A) is the probability of selecting someone with blue eyes. P(\bar{A}) is the probability of not selecting someone with blue eyes.

Step by step solution

01

- Understanding the Event A

First, understand that event A represents the event of selecting an adult with blue eyes. This means any probability related to event A will be connected to the proportion of adults with blue eyes.
02

- Define P(A)

The notation P(A) represents the probability of event A occurring. In this context, P(A) is the probability that a randomly selected adult will have blue eyes.
03

- Define \(P(\bar{A})\)

The notation \(P(\bar{A})\) represents the probability of event A not occurring. In other words, \(P(\bar{A})\) is the probability that a randomly selected adult will not have blue eyes.

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Key Concepts

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

event probability
Probability is a measure of how likely an event is to occur. When we deal with events in probability, such as selecting an adult with blue eyes, we often use specific notation to express these chances.

To denote the probability of an event, say event A, we use the notation \( P(A) \). This tells us the likelihood of event A happening. For example, if event A is selecting an adult with blue eyes, then \( P(A) \) gives us the chance or proportion of adults that have blue eyes.

Probability values range from 0 to 1, where 0 means the event cannot happen, and 1 means it will definitely occur. For instance:
  • \( P(A) = 0 \) implies no adults have blue eyes.
  • \( P(A) = 1 \) implies every adult has blue eyes.
  • \( P(A) = 0.5 \) implies half of the adults have blue eyes.

    • Understanding event probability is fundamental in statistics and helps us make predictions based on data.
eye color
Eye color is a genetic trait that can vary greatly from person to person. Common eye colors include brown, blue, green, and hazel. In our example, we are interested in the proportion of adults with blue eyes, represented by event A. Knowing the probability of selecting someone with blue eyes helps us understand its distribution in a given population.

Factors like genetics, geographical location, and demographic makeup influence the frequency of eye color. For instance:
  • In some regions, blue eyes are more prevalent, especially in Northern Europe.
  • In other regions, brown eyes might be more common.

    • When handling problems about traits like eye color, defining events clearly and knowing how to use probability notation can provide valuable insights into genetic and population studies.
complementary events
In probability, complementary events refer to a pair of outcomes that cover all possibilities. For an event A, its complement is noted as \( \bar{A} \). If event A is selecting an adult with blue eyes, then \( \bar{A} \) represents the event of selecting an adult without blue eyes.

The probabilities of an event and its complement always add up to 1, which means: \( P(A) + P(\bar{A}) = 1 \). This sum rule is because either the event happens, or it doesn鈥檛 鈥� there are no other possibilities.

Examples can help to clarify:

  • If \( P(A) = 0.3 \), meaning 30% of adults have blue eyes, then \( P(\bar{A}) \) must be 0.7 or 70%, representing those without blue eyes.

  • If \( P(A) = 0.6 \), then \( P(\bar{A}) \) will be 0.4.

    • Understanding complementary events is essential for solving many probability problems, as it lets us easily find the probability of an event not occurring.

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Most popular questions from this chapter

Find the probability and answer the questions. MicroSort's YSORT gender selection technique is designed to increase the likelihood that a baby will be a boy. At one point before clinical trials of the YSORT gender selection technique were discontinued, 291 births consisted of 239 baby boys and 52 baby girls (based on data from the Genetics \& IVF Institute). Based on these results, what is the probability of a boy born to a couple using MicroSort's YSORT method? Does it appear that the technique is effective in increasing the likelihood that a baby will be a boy?

Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Simulating Hybridization When Mendel conducted his famous hybridization experiments, he used peas with green pods and yellow pods. One experiment involved crossing peas in such a way that \(75 \%\) of the offspring peas were expected to have green pods, and \(25 \%\) of the offspring peas were expected to have yellow pods. Describe a procedure for using software or a TI-83/84 Plus calculator to simulate 20 peas in such a hybridization experiment. Each of the 20 individual outcomes should be an indication of one of two results: (1) The pod is green; (2) the pod is yellow.

Find the probability and answer the questions.Men have XY (or YX) chromosomes and women have XX chromosomes. X-linked recessive genetic diseases (such as juvenile retinoschisis) occur when there is a defective \(\mathrm{X}\) chromosome that occurs without a paired \(\mathrm{X}\) chromosome that is not defective. In the following, represent a defective \(X\) chromosome with lowercase \(x,\) so a child with the \(x\) Y or Yx pair of chromosomes will have the disease and a child with \(X X\) or XY or YX or \(x\) X or \(X x\) will not have the disease. Each parent contributes one of the chromosomes to the child.a. If a father has the defective \(x\) chromosome and the mother has good XX chromosomes, what is the probability that a son will inherit the disease? b. If a father has the defective \(x\) chromosome and the mother has good \(X X\) chromosomes, what is the probability that a daughter will inherit the disease? c. If a mother has one defective \(x\) chromosome and one good \(X\) chromosome and the father has good XY chromosomes, what is the probability that a son will inherit the disease? d. If a mother has one defective \(x\) chromosome and one good \(X\) chromosome and the father has good XY chromosomes, what is the probability that a daughter will inherit the disease?

Who discovered penicillin: Sean Penn, William Penn, Penn Jillette, Alexander Fleming, or Louis Pasteur? If you make a random guess for the answer to that question, what is the probability that your answer is the correct answer of Alexander Fleming?

Express the indicated degree of likelihood as a probability value between \(\boldsymbol{0}\) and \(\boldsymbol{I}\).Based on a report in Neurology magazine, \(29.2 \%\) of survey respondents have sleepwalked.
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