Chapter 20: Problem 12
When a coin is tossed, it can land heads up or tails up. You toss a coin 10 times, and it comes up heads every time. What is the probability that the coin will come up heads on the 11 th toss? a) \(10 \%\) c) 5096 e) \(100 \%\) b) \(20 \%\) d) \(90 \%\)
Short Answer
Expert verified
Answer: 50%
Step by step solution
01
Identify the probability of a single event (coin toss)
In each coin toss, there are two possible outcomes: heads or tails. Assuming the coin is fair, both outcomes are equally likely. Therefore, the probability of the coin landing heads up in a single toss is \(\frac{1}{2}\), since there is 1 desirable outcome (heads) out of a total of 2 possible outcomes (heads and tails).
02
Understand the independence of coin tosses
Coin tosses are independent events, meaning that the outcome of one toss does not affect the outcome of others. In this case, the fact that the coin landed heads up during the first 10 tosses has no influence on the outcome of the 11th toss. The probability of the coin landing heads up on the 11th toss remains the same as it was for each of the previous tosses.
03
Calculate the probability of the coin landing heads up on the 11th toss
Since the coin tosses are independent, the probability of the coin landing heads up on the 11th toss is the same as the probability of it landing heads up on any single toss. We found this probability in Step 1, which is \(\frac{1}{2}\). To express this as a percentage, we can multiply by 100: \(\frac{1}{2} \times 100 = 50 \%\).
04
Match the calculated probability to the given options
We calculated the probability of the coin landing heads up on the 11th toss to be \(50 \%\). Comparing our result with the given options, we can conclude that none of the options match our calculated probability. There must be an error in the given options, since the correct answer should be \(50 \%\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coin Toss Probability
Understanding the concept of coin toss probability is fundamental to grasping many basic principles of probability and statistics. A coin toss is a classic example of a random experiment with two possible outcomes, often labeled as heads (H) and tails (T). Assuming that the coin is unbiased and fair, each outcome has an equal chance of occurring on any given toss.
When you flip a fair coin once, there is a 1 in 2 chance, or a probability of \(\frac{1}{2}\), that it will land heads up. This same probability applies to each subsequent coin toss, regardless of previous results. This is because the coin does not have any memory of past flips; hence, each toss is a separate event with the same likelihood of resulting in heads or tails.
In our case, even after observing 10 consecutive heads, the 11th coin toss maintains a probability of \(\frac{1}{2}\) for landing heads up. This is contrary to the common misconception that previous outcomes can influence future events, a fallacy known as the gambler's fallacy.
When you flip a fair coin once, there is a 1 in 2 chance, or a probability of \(\frac{1}{2}\), that it will land heads up. This same probability applies to each subsequent coin toss, regardless of previous results. This is because the coin does not have any memory of past flips; hence, each toss is a separate event with the same likelihood of resulting in heads or tails.
In our case, even after observing 10 consecutive heads, the 11th coin toss maintains a probability of \(\frac{1}{2}\) for landing heads up. This is contrary to the common misconception that previous outcomes can influence future events, a fallacy known as the gambler's fallacy.
Independence in Probability
Independence in probability is a concept that states that the occurrence of one event does not affect the probability of another event occurring. In simpler terms, two events are independent if knowing the outcome of one provides no information about the outcome of the other. This is an essential concept to understand when calculating probabilities in various scenarios.
For example, coin tosses are independent events because the result of one toss does not influence or change the chances of the result in another toss. Mathematically, if events A and B are independent, the probability of both events occurring is the product of their individual probabilities: \( P(A \text{ and } B) = P(A) \times P(B)\).
Applying this to multiple coin tosses, the outcome of the 11th toss is independent of the first 10 tosses. Consequently, each toss has a 50% chance of being heads, and past tosses do not change this. Thus, a series of tosses can result in any combination of heads and tails, with the probability of heads on any given toss always remaining at \(\frac{1}{2}\).
For example, coin tosses are independent events because the result of one toss does not influence or change the chances of the result in another toss. Mathematically, if events A and B are independent, the probability of both events occurring is the product of their individual probabilities: \( P(A \text{ and } B) = P(A) \times P(B)\).
Applying this to multiple coin tosses, the outcome of the 11th toss is independent of the first 10 tosses. Consequently, each toss has a 50% chance of being heads, and past tosses do not change this. Thus, a series of tosses can result in any combination of heads and tails, with the probability of heads on any given toss always remaining at \(\frac{1}{2}\).
Basic Statistics
Basic statistics involve collecting, analyzing, interpreting, presenting, and organizing data. It provides tools for making sense of data and making informed decisions based on evidence. Probability plays a significant role in statistics, as it allows us to quantify uncertainty and measure the likelihood of various outcomes.
In the realm of basic statistics, understanding probability distributions is crucial. For a fair coin, the distribution of landing on heads or tails follows a binomial distribution, where each toss is independent, and the probability for each outcome (heads or tails) remains constant at \(\frac{1}{2}\). Furthermore, in basic statistical analysis, it is important to recognize the difference between dependent and independent events, as we've seen with the independence of coin tosses.
Moreover, in basic statistics, it's also important to avoid common pitfalls such as the aforementioned gambler's fallacy or misinterpreting the Law of Large Numbers, which states that as the number of trials increases, the empirical probability (observed outcomes) will tend to approach the theoretical probability of an event.
In the realm of basic statistics, understanding probability distributions is crucial. For a fair coin, the distribution of landing on heads or tails follows a binomial distribution, where each toss is independent, and the probability for each outcome (heads or tails) remains constant at \(\frac{1}{2}\). Furthermore, in basic statistical analysis, it is important to recognize the difference between dependent and independent events, as we've seen with the independence of coin tosses.
Moreover, in basic statistics, it's also important to avoid common pitfalls such as the aforementioned gambler's fallacy or misinterpreting the Law of Large Numbers, which states that as the number of trials increases, the empirical probability (observed outcomes) will tend to approach the theoretical probability of an event.
