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Chapter 12: Problem 56

Nickels falling over. You may feel that it is obvious that the probability of a head in tossing a coin is about \(1 / 2\) because the coin has two faces. Such opinions are not always correct. Stand a nickel on edge on a hard, flat surface. Pound the surface with your hand so that the nickel falls over. What is the probability that it falls with heads upward? Make at least 50 trials to estimate the probability of a head.

Short Answer

Expert verified
Estimate the probability by dividing heads outcomes by 50.

Step by step solution

01

Conduct the Experiment

Take a nickel, stand it on its edge on a hard, flat surface, and then cause it to fall by pounding the surface. Record whether the nickel lands with heads or tails facing up. Repeat this process for a total of 50 trials to collect data on how many times the nickel lands on heads versus tails.
02

Record and Count Outcomes

For each trial, note the outcome as heads or tails. Create a table or list to keep track of how many times the nickel lands on heads throughout the 50 trials. Ensure the total adds up to 50, which is the number of trials conducted.
03

Calculate the Probability of Heads

To find the probability of the nickel landing heads up, take the number of times it landed on heads and divide by the total number of trials. Use the formula: \( P(\text{heads}) = \frac{\text{Number of heads}}{\text{Total trials}} \). For example, if the nickel lands on heads 20 out of 50 times, the probability would be \( \frac{20}{50} = 0.4 \) or 40%.

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Key Concepts

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

Probability Estimation
Probability estimation is a crucial aspect of understanding the likelihood of a particular outcome occurring in an experiment. In the case of the nickel experiment, we're trying to estimate how often the nickel lands on heads when it is knocked over. To do this, we rely on data gathered from multiple observations or trials. By conducting 50 trials, we aim to gather enough information to approximate the chance of this outcome.

Here's how probability estimation works:
  • Each time you conduct a trial where the nickel is knocked over, you are gathering one piece of data.
  • The more trials you conduct, the more accurate your probability estimation is likely to be.
  • By dividing the number of successful trials (nickel landing on heads) by the total number of trials conducted, you get an estimated probability.
Such estimation helps us make predictions. However, remember that the estimation can vary. It may change slightly every time you repeat the experiment. Thus, having numerous trials helps stabilize the probability estimation, making it a more reliable representation of the nickel's behavior in this experiment.
Statistical Trials
Statistical trials are repetitions of an experiment used to observe outcomes and, subsequently, gather meaningful data. In the context of the nickel experiment, each time the nickel is made to fall, we conduct a single trial. This process is repeated multiple times鈥攊n this example, 50 times鈥攖o build up a data set.

Why are statistical trials important?
  • They provide a structured way of observing and recording outcomes.
  • The repetition reduces the impact of any anomalies or outliers, leading to a more stable result.
  • The results from many trials can be combined to calculate experimental probabilities.
In essence, the more trials you conduct, the more trustworthy your findings. Particularly in probability and statistics, having a large number of trials smooths out random variations in the data. This ensures the results reflect an accurate depiction of the probability of specific outcomes.
Experimental Probability
Experimental probability plays a key role in understanding random events based on actual experiments. It allows us to determine the likelihood of an event occurring by using direct measurements and observations, rather than relying solely on theoretical probability. In the nickel experiment, experimental probability gives insight into how often heads appears by using the results of physical trials.

The calculation involved is straightforward:
  • Count how many times heads appears (successful trials).
  • Divide this number by the total number of trials to get a probability estimation.
Unlike theoretical probability, which might suggest a 50% chance for heads due to two possible faces, experimental probability could reveal a different percentage based on observations. This makes experimental probability particularly beneficial in scenarios where theoretical assumptions do not easily apply, highlighting real-world variations and nuances that theoretical models might not predict.

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Most popular questions from this chapter

The Medical College Admission Test. The Normal distribution with mean \(\mu=499.6\) and standard deviation \(\sigma=10.4\) is a good description of the total score on the Medical College Admission Test (MCAT). \({ }^{9}\) This is a continuous probability model for the score of a randomly chosen student. Call the score of a randomly chosen srudent \(X\) for short. (a) Write the event "the student chosen has a score of 510 or higher" in terms of \(\boldsymbol{K}\). (b) Find the probability of this event.

A taste test. A tea-drinking Canadian friend of yours claims to have a very refined palate. She tells you that she can tell if, in preparing a cup of tea, milk is first added to the cup and then hot tea poured into the cup, or the hot tea is first poured into the cup and then the milk is added. \({ }^{18}\) To test her claims, you prepare six cups of tea. Three have the milk added first and the other three the tea first. In a blind taste test, your friend tastes all six cups and is asked to identify the three that had the milk added first. (a) How many different ways are there to select three of the six cups? (Hint: See Example 12.8.) (b) If your friend is just guessing, what is the probability that she correctly identifies the three cups with the milk added first?

A door prize. A party host gives a door prize to one guest chosen at random. There are 48 men and 42 women at the party. What is the probability that the prize goes to a woman? Explain how you arrived at your answer.

You read in a book on poker that the probability of being dealt a straight flush in a five-card poker hand is \(1 / 64,974\). This means that (a) if you deal millions of poker hands, the fraction of them that contain a straight flush will be very close to \(1 / 64,974 .\) (b) if you deal 64,974 poker hands, exactly one of them will contain a straight flush. (c) if you deal \(6,497,400\) poker hands, exactly 100 of them will contain a straight flush.

Running a Mile. A study of 12,000 able-bodied male students at the University of Illinois found that their times for the mile run were approximately Normal with mean \(7.11\) minutes and standard deviation \(0.74\) minute. \({ }^{11}\) Choose a student at random from this group and call his time for the mile \(Y\). (a) Is \(Y\) a finite or continuous random variable? Explain your answer. (b) Say in words what the meaning of \(P(Y \geq 3.0)\) is. What is this probability? (c) Write the event "the student could run a mile in less than 6 minutes" in terms of values of the random variable \(Y\). What is the probability of this event?
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