Understanding Probability of Bias in Coin and Dice Tosses

In summary, the conversation discusses the concept of calculating the probability of a coin or die being loaded or biased. It is stated that this probability cannot be determined without a prior probability of the coin or die being biased. The conversation also touches on the use of Bayesian statistics in this scenario.
  • #1
bsharvy
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TL;DR Summary
Can you find the probability that a coin or die is loaded?
I was thinking that the probability of a set of events not happening is the same as the probability of that the die/coin is biased.

So, if I flip a coin 10 times and get heads every time, the probability the coin is biased is 1- (.5)^7.

Roll a die 5 times, get "4" all times, probability of bias = 1 - (1/6)^5

But that suggests the probability of bias after one coin toss is 50%, which can't be right. I'm also not sure how to calculate when the results are mixed, such as flipping a coin 10 times and getting heads 7 times.

Help!
 
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  • #2
Your question has a subtle, but important, mistake. Whether a coin is biased is not a random variable with a probability. So you should not talk about its probability. The only probability that you can talk about is that of getting a certain result from a fair coin. If that probability is too low, you are justified in rejecting the assumption that the coin is fair. This is called "rejecting the null hypothesis". There are certain standard levels of small probability, called "significance levels", that allow you to reject the null hypothesis. They include 0.05, 0.025, and 0.01. These are the probabilities of rejecting the null hypothesis even though it is true, so smaller is better. For something like the discovery of a new elementary particle in science, the significance level required is extremely demanding. It is "5 sigma", which is a probability of ##3\times 10^{-7}## or about one in 3.5 million.
 
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  • #3
bsharvy said:
TL;DR Summary: Can you find the probability that a coin or die is loaded?

I was thinking that the probability of a set of events not happening is the same as the probability of that the die/coin is biased.

So, if I flip a coin 10 times and get heads every time, the probability the coin is biased is 1- (.5)^7.

Roll a die 5 times, get "4" all times, probability of bias = 1 - (1/6)^5

But that suggests the probability of bias after one coin toss is 50%, which can't be right. I'm also not sure how to calculate when the results are mixed, such as flipping a coin 10 times and getting heads 7 times.

Help!
You cannot compute the probability that a coin is biased without a prior probability of the coin being biased. Consider the two extreme cases:

1) There are no biased coins.

2) All coins are biased.

To take a more realistic example: assume that one coin in a hundred is biased (and to keep things simple comes up heades every times). Out of approximately every 1100 coins we have:

1000 fair coins that give ten heads in a row approximately once;
11 biased coins that come up heads every time.

If we get 10 heads in a row, then the probability is about 11/12 that the coin is biased.

If we assume that one coin in a thousand or one coin in a million is biased, we will get very different answers.
 
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  • #4
FactChecker said:
Whether a coin is biased is not a random variable with a probability. So you should not talk about its probability.
It certainly can be considered a random variable with a probability distribution in Bayesian statistics. But in frequentist statistics it would not be a random variable.

PeroK said:
You cannot compute the probability that a coin is biased without a prior probability of the coin being biased.
Yes, I agree, the prior probability is needed.

@bsharvy basically this is an exercise in Bayesian statistics. I have a set of Insights articles on this topic. In particular, the third one may be useful for you since I specifically examine the fair coin question in some depth:

https://www.physicsforums.com/insights/how-bayesian-inference-works-in-the-context-of-science/
 
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FAQ: Understanding Probability of Bias in Coin and Dice Tosses

What is probability bias in coin and dice tosses?

Probability bias refers to the tendency for a certain outcome to occur more frequently than others in a random event. In the case of coin and dice tosses, this means that one side of the coin or one number on the dice is more likely to appear than others due to external factors such as the weight or shape of the coin/dice, or the way it is tossed.

How does probability bias affect the outcome of coin and dice tosses?

Probability bias can significantly impact the outcome of coin and dice tosses. It can make certain outcomes more likely to occur, while decreasing the chances of others. This can result in an unfair or skewed representation of the true probability of each outcome.

What are some common examples of probability bias in coin and dice tosses?

Some common examples of probability bias in coin and dice tosses include a weighted coin, where one side is slightly heavier than the other, or a loaded die, where one number is more likely to appear due to its placement or added weight. Other factors such as the surface the coin/dice is being tossed on or the force used to toss it can also contribute to bias.

How can we identify and prevent probability bias in coin and dice tosses?

To identify and prevent probability bias in coin and dice tosses, it is important to use a fair and standardized method of tossing, such as flipping the coin with a flat hand or rolling the dice in a cup and releasing them onto a flat surface. It is also important to use unbiased coins and dice, and to rotate them regularly to prevent wear and tear that could lead to bias.

Why is understanding probability bias important in scientific research?

Understanding probability bias is crucial in scientific research as it helps to ensure the accuracy and reliability of results. If bias is present in the method of data collection, it can lead to incorrect conclusions and flawed research. By recognizing and accounting for probability bias, scientists can ensure that their experiments and studies are truly representative of the underlying probabilities and produce valid and trustworthy results.

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