Unraveling Probability: Coin Flips, Clinical Trials, and Conspiracy Theories!

Have you ever found yourself puzzled by extraordinary claims about alternative cures or controversies around vaccines despite overwhelming scientific evidence to the contrary? In our interconnected world, these tales spread rapidly, and a seemingly rare occurrence can get amplified disproportionately. What many overlook is the key role of a misunderstood mathematical principle: Probability, particularly when we deal with large numbers.

In a world of billions, the rarest of events are bound to happen. What may appear extraordinary on an individual level is statistically expected, resulting in a skewed perception of what is “likely.” In this article, we’ll take a deep dive into the fascinating world of probability and how misinterpretations can influence public opinion, particularly during events like a global pandemic.

We’ll use the simple, yet powerful analogy of flipping a coin in a packed stadium of 10,000 people, demonstrating how even the most unlikely events become near certainties when performed a large number of times. This will provide insights into real-world scenarios like clinical trials and health misinformation.

Join us as we unravel the math behind these phenomena, and why, in a world with billions of people, we should be prepared for the unexpected — because statistically speaking, the unexpected is expected. By understanding this, we can better differentiate between genuine patterns and random noise in our world of information overload.

Understanding Multiple Testing Errors:

Multiple testing errors occur when researchers perform several statistical tests on the same dataset, which increases the likelihood of finding a significant result by chance alone. This can lead to false positives (Type I errors) and false negatives (Type II errors). False positives are when a test incorrectly indicates a significant effect, while false negatives occur when a test fails to detect a genuine effect.

Flipping a fair coin 10 times has 2¹⁰ (1,024) possible outcomes because each of the 10 flips has 2 possibilities with an expected outcome of 50% heads and 50% tails. This means the probability of getting exactly five heads and five tails is probably far lower than you would intuitively expect. We can use the binomial coefficient formula to calculate how many of those 1,024 possible outcomes are exactly 5 heads and 5 tails. The binomial coefficient formula is:

C(n, k) = n! / (k!(n-k)!)

Where n is the number of flips and k is the desired number of heads (or tails since the probabities of both are equal). In our case, n = 10 (flips) and k = 5 (desired heads) and the formula tells us that there are 252 ways to get exactly five heads and five tails in 10 flips, a probability of approximately 24.6%.

Probability = 252 / 1,024 ≈ 0.246 (24.6%)

Thus, the probability of getting exactly five heads and five tails in 10 flips is about 24.6%. This means that there is a 75.4% chance of not getting five heads and five tails, even though the expected outcome is 50% heads and 50% tails for a fair coin flip. Using the same probability formula we see that the chances of flipping 10 of the same face (heads or tails) is 0.1953% (or a probability of 0.00195) . Thats shouldn’t be surprising becasue most people can intuitively deduce that the probabiliy is probably very low.

Now, picture a stadium with 10,000 people, each flipping a coin 10 times. What are the chances that at least one person gets 10 of the same flip in a row? Since the sum of all probabilities for an outcome must be equal to 1 , the probability of a single event NOT happing is just 1 minus the probability of that single event happening . This is known as the “complementary probability”.

So what is the opposite of flipping 10 heads or tails in a row? Well … its flipping anything OTHER than 10 in a row. So if there is a 0.001953 probability of 10 identical flips, then the probability of all other outcomes combined must be 1–0.00195= 0.998 . In other words , there is a 99.8% chance that you will flip some mixture of heads and tails rather than all of one face . Again… this should be intuitive… no surprises yet.

But here is where the math begins to challenge your intuition . From a probability standpoint, our stadium of 10,000 coin flippers is no different from having one person do 10 flips … 10,000 times . The first flip does not influece the outcome of the 10,000th flip so we say that all outcomes are independent of each other .In cases like this , the probability of multiple events happening together is simpy the product of each event happening individually . So if there is a 0.9980 probability of NOT getting 10 in a row , then the probability of not getting 10 in a row twice is 0.9980² which is 0.9960… wait… do you see what happened there ? The chances of not flipping 10 in a row just decreased when we added the second flip. This makes sense though right ? You now have 2 chances to get 10 in a row . That is intuitively going to be a higher probability than if you had 1 chance . So what happens when we have 10,000 chances ? Its 0.9980¹⁰⁰⁰⁰ which is 0.000000002020286 ! There is a near mathematical certainty that there will be at least one flip that has 10 in a row. In fact, we could use the same complimentary probability formula and calculate that there is a greater than 99.999% chance that at least one person will flip 10 in a row. Remember that we are still talking about a coin flip that has a 50% chance of being either heads or tails .

So how many of our 10,000 flippers would see either 10 heads or 10 tails ? Thats simply the probability of that outcome happening once multiplied by the number of attempts . Earlier we calculated that probability as 0.00195 or 0.195%.

The number of people we would expect 0.00195*10,000 = 19.5 . Thats right , we fully expect to see about 20 people flip 10 heads or 10 tails using a coin that is 50/50 .

This example highlights how even extremely unlikely events can become statistical certainties in large populations. In clinical trials, multiple testing errors can lead to false positives or negatives, distorting our perception of a treatment’s effectiveness. Researchers can sometimes do everything correctly and still, by random luck, get a result that is unreliable . In our scenario of 10,000 coin flippers , each participant only makes one attempt . Each of the 20 individuals that saw 10 identical flips did so on their first try ! Furthermore , the roughly 75% of people that will see something other than a perfect 5/5 split also did so on their first try . To emphasize this point even further … there were 7,500 people in a group of 10,000 that have never actually seen a set of coin flips result in a 50/50 split even though they were flipping coins that had a 50% chance of landing on either side.

Now imagine that those people developed an entire view of the fairness of coin flips based on not only their personal experience , but also based on the conversations they had with the other 10,000 participants . There would be 2,500 people claiming that a fair coin has a 50% chance of landing on either side and there would be 7,5000 people saying its something other than 50/50 based on the experience they had . There would even be a small group of 10 individuals that firmly believe that coins only land on heads and an opposing group of 10 people that believe coins only land on tails . I can see the headlines now : “ Coin Flips are a Conspiracy!” . Only an observer who has knowledge of the entire dataset and can appropriately apply the necessary population statistics would see that the coins are indeed fair and what those individuals are experiencing is the natural probability distribution that we would expect from a coin flip.

Extrapolating to the Global Population:

Considering a global population of 8 billion people, we can further explore the impact of multiple testing errors. With such a large number, even extremely rare events become almost certain to occur. For example, using the same coin flip analogy and probability calculation, in a stadium with 8 billion people , we would expect approximately 15.6 million people to get either 10 heads or 10 tails in 10 flips. This striking figure demonstrates how even a seemingly rare outcome can become quite prevalent when we consider the vast scale of the world’s population. It’s important to remember that we are still discussing fair coin flips with an expected outcome of 50% heads and 50% tails, yet a staggering 15.6 million people globally experience the polar opposite of this expectation. Imagine now that these 15.6 million people , who are actually two distinct groups equally split between the people who saw 10 heads and those who saw 10 tails , form online communities to discuss and debate their experiences . Imagine the people who have never even flipped a coin reading this and determining that 15.6 million people’s very real shared expereinces must be indicative of some hidden truth about coin flips . Its very easy to see how mutliple testing errors can be problematic if not accounted for in your analysis right ?

The Impact on Public Perception:

The large numbers involved in global clinical trials or health-related events can lead to the formation of groups that share seemingly rare experiences. These groups, such as those advocating for alternative medical treatments or conspiracy theories, may genuinely believe their experiences represent a broader trend. They did afterall have very real and valid experiences that just happened to fall on the extreme ends of a global probability curve.

For example, consider the anti-vaccine movement, wherein some individuals argue that vaccines cause harm to their children, despite scientific data indicating otherwise. It is possible that the individuals claiming negative effects from vaccines are among the small fraction of people who experienced an unlikely outcome, akin to the 15.6 million people in our coin flip analogy.

We could similarly scrutinize the controversy surrounding the use of ivermectin as a potential treatment for COVID-19. Some small-scale clinical trials suggested that ivermectin could be effective, leading to widespread support for the drug among certain groups. However, when examined in the context of multiple testing errors and the vast scale of the pandemic, it becomes clear that these small-scale trials may not provide sufficient evidence for its effectiveness. Did some people take ivermectin and see an improvement ? Sure , just like some people flipped 10 coins and got 10 heads in our stadium example.

These groups may congregate online, amplifying their shared experiences and reinforcing their beliefs. This can lead to the spread of misinformation and potentially harmful ideas, as their experiences are interpreted as evidence against established scientific knowledge.

The Importance of Recognizing Multiple Testing Errors and Statistical Flukes:

Being aware of the role large numbers play in our perception of probability and its potential impact on public opinion is essential. Recognizing that some outcomes may result from multiple testing errors and chance alone allows us to better differentiate genuine patterns from random noise.

In the context of clinical trials and health-related events, relying on rigorous scientific research and established methodologies is crucial for determining treatment and intervention effectiveness. While individual experiences can provide valuable insights, they must be considered within the broader context of scientific evidence and potential multiple testing errors.

Conclusion:

Our intuition can deceive us when dealing with probabilities and large numbers, leading to misconceptions about the significance of certain events. By employing a coin flip analogy and real-life examples, we have demonstrated that even unlikely events can become statistical certainties in large populations, leading to the formation of groups with shared experiences contrary to scientific evidence. In a world with billions of people, it is vital to separate genuine patterns from random noise, multiple testing errors, and rely on established scientific knowledge to inform our decisions about health and well-being.

This is particularly important during global events like the pandemic, where the sheer scale of the population can make multiple testing errors and misinterpretations of data more prevalent. As we continue to navigate these challenges, remaining vigilant and relying on rigorous scientific research and established methodologies will be crucial in determining the effectiveness of treatments and interventions, ultimately contributing to better health outcomes for everyone.