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He cheats? A friend of yours claims that when he tosses a coin he can control the outcome. You are skeptical and want him to prove it. He tosses the coin, and you call heads; it's tails. You try again and lose again. a. Do two losses in a row convince you that he really can control the toss? Explain. b. You try a third time, and again you lose. What's the probability of losing three tosses in a row if the process is fair? c. Would three losses in a row convince you that your friend controls the outcome? Explain. d. How many times in a row would you have to lose to be pretty sure that this friend really can control the toss? Justify your answer by calculating a probability and explaining what it means.

Step by step solution

01

Interpret the Experiment

Firstly, when a fair coin is tossed, both outcomes (Heads and Tails) are equally likely. Each toss is an independent event, so the outcome of one does not affect the outcome of another.

02

Calculate Probability of Two Losses

In this case, a 'loss' is getting tails. The probability of getting tails in each toss is \(\frac{1}{2} = 0.5\) because there are 2 equally likely outcomes. Therefore, the chances of losing twice in a row is \(0.5 * 0.5 = 0.25\) or 25%.

03

Calculate Probability of Three Losses

Following the same logic, losing a third time equates to getting tails again, which also has a probability of 0.5. Thus, the probability of losing three times in a row is \(0.5 * 0.5 * 0.5 = 0.125\) or 12.5%.

04

Calculate Arbitrary Number of Losses

To calculate the probability of losing 'x' times in a row, you would have to multiply 0.5 by itself 'x' times. This result gets smaller the larger 'x' gets, thus the less likely the outcome is just by chance and the more indication there is of some control over the toss.

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Key Concepts

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

Independent Events

When studying independent events, it's important to realize that these events do not affect one another.
In the context of a fair coin toss, each flip is an independent event. This means the result of one toss—whether it is heads or tails—does not influence the outcome of the subsequent toss.
An easy way to think about this is by considering that every time a coin is tossed, it's like starting fresh.

• If you toss a coin and get heads, the probability of getting heads or tails on your next toss still remains unaffected at 50% for each outcome.
• This property of independence in probability is a crucial aspect of random experiments, emphasizing the chance-driven nature of each event.

Understanding this concept helps us analyze situations without falsely attributing a pattern or influence to random events.

Fair Coin Toss

A fair coin toss implies that the coin is unbiased, meaning it doesn't favor heads or tails.
Essentially, each outcome has an equal probability of occurring, which is \(\frac{1}{2}\) or 50% for heads and 50% for tails.
To better comprehend this concept:

• Think of a perfect balance, like a scale, where each side is equally weighted.
  This balanced aspect ensures that no external factors or imperfections in the coin introduce a bias.
• A fair coin creates a simple model of probability where outcomes are predictably random and each trial is unaffected by previously recorded results, aligning well with the idea of independent events.

In scenarios where fairness is assumed, such as coin flipping games or probability exercises, it serves as a basis for understanding more complex probability calculations.

Probability Calculation

Probability calculation involves determining the likelihood of a specific outcome or a series of outcomes.
For a fair coin toss, calculating probabilities also demonstrates how different events interact when they are combined.
Here's how to calculate it step-by-step:

• Start by acknowledging that each individual toss has a probability of \(\frac{1}{2}\) for either outcome.
• When you want to calculate the probability of multiple outcomes in a row (like losing three times or "tails" each time), you multiply the probability of a single event occurring by itself for each event.
  So, for three tails in a row, it's \(0.5 \times 0.5 \times 0.5 = 0.125\) which translates to 12.5%.
• This multiplication rule applies because each toss is independent, and thus we repeatedly apply the basic probability for each additional event.

Understanding how to calculate these probabilities is essential in determining how rare or expected a particular sequence of outcomes is under given conditions.