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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

Sample spaces For each of the following, list the sample space and tell whether you think the events are equally likely: a. Toss 2 coins; record the order of heads and tails. b. A family has 3 children; record the number of boys. c. Flip a coin until you get a head or 3 consecutive tails; record each flip. d. Roll two dice; record the larger number (or simply the number in case of a tie).

Blood The American Red Cross says that about \(45 \%\) of the U.S. population has Type O blood, \(40 \%\) Type A, \(11 \%\) Type B, and the rest Type \(\mathrm{AB}\). a. Someone volunteers to give blood. What is the probability that this donor 1\. has Type AB blood? 2\. has Type A or Type B? 3\. is not Type O? b. Among four potential donors, what is the probability that 1\. all are Type O? 2\. no one is Type AB? 3\. they are not all Type \(\mathrm{A}\) ? 4\. at least one person is Type B?

Dice You roll a fair die three times. What is the probability that a. you roll all 6 's? b. you roll all odd numbers? C. none of your rolls gets a number divisible by \(3 ?\) d. you roll at least one \(5 ?\) e. the numbers you roll are not all 5's?

Polling, part II According to Pew Research, the contact rate (probability of contacting a selected household) was \(90 \%\) in 1997 and \(62 \%\) in 2012 . However, the cooperation rate (probability of someone at the contacted household agreeing to be interviewed) was \(43 \%\) in 1997 and dropped to \(14 \%\) in \(2012 .\) a. What is the probability (in 2012 ) of obtaining an interview with the next household on the sample list? (To obtain an interview, an interviewer must both contact the household and then get agreement for the interview.) b. Was it more likely to obtain an interview from a randomly selected household in 1997 or in \(2012 ?\)

Lefties Although it's hard to be definitive in classifying people as right- or left-handed, some studies suggest that about \(14 \%\) of people are left- handed. Since \(0.14 \times 0.14=0.0196,\) the Multiplication Rule might suggest that there's about a \(2 \%\) chance that a brother and a sister are both lefties. What's wrong with that reasoning?
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