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Chapter 12: Problem 1

Flipping a coin Flipping a fair coin is said to randomly generate heads and tails with equal probability. Explain what random means in this context.

Short Answer

Expert verified
In the context of flipping a fair coin, random means that each flip is an independent event with no influence on or from other flips and has an equal chance (50%) of landing on either heads or tails.

Step by step solution

01

Understanding concept of random

Random refers to the concept that each event in a series of events happens independently, without any pattern or predictability. Every event or instance does not influence or have a connection to the previous or subsequent occurrence.
02

Understanding concept of equal probability

Equal probability means any of the outcomes has the same chance of occurring. In this context, it implies that each time a fair coin is flipped, it has the same chance of showing heads or tails.
03

Applying concepts to context of the fair coin

In the context of flipping a fair coin, random means that any series of flips (heads or tails) does not follow any predictable pattern and each flip is an independent event, and has an equal probability i.e., there's a 50% chance it will land heads up and a 50% chance it will land tails up.

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Key Concepts

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

Randomness
In the realm of probability, randomness plays a crucial role. When we mention that something is random, we are indicating that each outcome is unpredictable and not influenced by any previous events. Take the flipping of a coin, for instance. If you flip a coin multiple times, each flip does not possess any memory of what happened before it.
No matter how many times you land on heads in a row, the next flip still holds an equal chance of showing heads or tails. This lack of predictability and pattern is essentially randomness at work. It's what gives the process its fairness, as each possible result stands independently of any previous outcomes.
Independence
Independence in probability refers to the scenario where one event does not affect the outcome of another event. In the context of flipping a coin, each flip is completely independent of another.
This means that the outcome of your previous flip, whether it shows heads or tails, has zero influence over the result of your upcoming flip. Consequently, even if you have flipped five heads in a row, the next flip's probability remains unchanged.
  • Each flip is isolated.
  • Previous results do not alter the probabilities.
  • Independence ensures fairness and consistency.
Understanding this concept is vital to grasping why each coin toss is a fresh and independent event.
Fair Coin
A fair coin is a central concept in the study of probability. By 'fair,' we mean that the coin is perfectly balanced, and there is no weight, shape, or any other factor that biases the outcome toward heads or tails.
When you toss a fair coin, you can expect it to not favor one side over the other. This characteristic ensures that each side has an equal opportunity, providing the bases of fair probability analysis. An unreliable coin 鈥� perhaps one that is weighted incorrectly 鈥� can skew results, making one side more likely to appear more often, thus defeating the fairness in randomness.
  • No biases with a fair coin.
  • Balance is key to fairness.
  • Ensures true randomness in results.
Consequently, using a fair coin is essential for accurate probability experiments.
Equal Probability
Equal probability is a fundamental principle of probability, particularly in random events like coin flips. It means each outcome in a given event has the same likelihood of occurring. For a fair coin, this means there is a 50% chance for it to land on heads and a 50% chance to land on tails.
This balance is essential to fairness because it ensures no particular outcome is more favored than another.
  • Each outcome has an equal chance.
  • For a coin: 50% heads, 50% tails.
  • Ensures unbiased results.
In probability, equal probability enables a level playing field, which is a cornerstone of random processes.

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Most popular questions from this chapter

Roulette A casino claims that its roulette wheel is truly random. What should that claim mean?

Red cards You shuffle a deck of cards and then start turning them over one at a time. The first one is red. So is the second. And the third. In fact, you are surprised to get 10 red cards in a row. You start thinking, "The next one is due to be black!" a. Are you correct in thinking that there's a higher probability that the next card will be black than red? Red cards You shuffle a deck of cards and then start turning them over one at a time. The first one is red. So is the second. And the third. In fact, you are surprised to get 10 red cards in a row. You start thinking, "The next one is due to be black!" a. Are you correct in thinking that there's a higher probability that the next card will be black than red?

Lights You purchased a five-pack of new light bulbs that were recalled because \(6 \%\) of the lights did not work. What is the probability that at least one of your lights is defective?

Cell phones and surveys A 2010 study conducted by the National Center for Health Statistics found that \(25 \%\) of U.S. households had no landline service. This raises concerns about the accuracy of certain surveys, as they depend on random-digit dialing to households via landlines. We are going to pick five U.S. households at random: a. What is the probability that all five of them have a landline? b. What is the probability that at least one of them does not have a landline? c. What is the probability that at least one of them does have a landline?

College admissions For high school students graduating in 2013, college admissions to the nation's most selective schools were the most competitive in memory. Harvard accepted about \(5.8 \%\) of its applicants, Dartmouth \(10 \%,\) and Penn \(12.1 \%\). Jorge has applied to all three. Assuming that he's a typical applicant, he figures that his chances of getting into both Harvard and Dartmouth must be about \(0.58 \%\). a. How has he arrived at this conclusion? b. What additional assumption is he making? c. Do you agree with his conclusion?
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