When the Wright brothers were deciding who would be the first to fly their new airplane, they used a simple coin flip. This was fair because there’s an equal chance of landing on heads or tails. But what if their contest was more complicated? Imagine they flipped coins repeatedly, with Orville winning if two heads appeared in a row, and Wilbur winning if heads was followed by tails. Would they both have the same chance of winning?
At first glance, it might seem like they would have equal chances since there are four possible outcomes for two flips, each with a 25% chance. However, this assumption is incorrect. In reality, Wilbur has a better chance of winning in this scenario.
To understand why Wilbur has an advantage, think of the coin flips as a board game. Each flip determines the path you take, and the goal is to reach the finish line. The path for heads followed by tails (heads/tails) and the path for two heads in a row (heads/heads) are different. The heads/heads path has a move that can send you back to the start, making it take longer on average compared to the heads/tails path.
We can use probability and algebra to figure out the average number of flips needed for each scenario. For the heads/tails path, let’s call the average number of flips to advance one step “x.” If you flip tails, you stay in place and need to flip again, averaging x + 1 flips. If you flip heads, you advance in one flip. Combining these possibilities, it takes an average of two flips to advance one step, and four flips to advance two steps.
For the heads/heads path, let’s call the average number of flips to finish “y.” If you flip tails first, you go back to the start, averaging y + 1 flips. If you flip heads first, and then another heads, you finish in two flips. But if the second flip is tails, you return to the start, needing an average of y + 2 flips. Solving this, it takes an average of six flips for heads/heads and four flips for heads/tails.
This means that achieving heads/heads takes longer than heads/tails. In reality, the Wright brothers only flipped the coin once, and Wilbur won. However, his flight didn’t succeed, and it was Orville who made aviation history.
This coin flip conundrum teaches us about probability and how outcomes can be less obvious than they seem. Even simple games can have surprising results when you dig into the math behind them!
Gather in pairs and simulate the coin flip game described in the article. Use a real coin and keep track of the results. One student plays as Orville, winning with two heads in a row, and the other as Wilbur, winning with heads followed by tails. Record the number of flips needed for each win and compare the results with the theoretical probabilities discussed in the article.
Draw a probability tree diagram to visualize the possible outcomes of two consecutive coin flips. Label each branch with the probability of that outcome. Use this diagram to calculate the probability of each player winning and discuss how this visual representation helps in understanding the problem.
Use a simple computer program or an online simulation tool to model the coin flip game. Run the simulation multiple times to see how often each player wins. Compare these results with your manual simulations and the theoretical probabilities. Discuss any differences you observe and what might cause them.
In small groups, discuss other real-life situations where probability plays a crucial role. Consider games, decision-making processes, or scientific experiments. Share your findings with the class and relate them back to the coin flip conundrum to deepen your understanding of probability.
Write a short reflection on what you learned about probability from the coin flip conundrum. Consider how your understanding of probability has changed and how you might apply this knowledge in the future. Share your reflections with a partner and discuss any new insights you gained.
Here’s a sanitized version of the transcript, removing any unnecessary details while retaining the core concepts:
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When the Wright brothers needed to decide who would be the first to fly their new airplane off a sand dune, they flipped a coin. This was a fair method, as there’s an equal chance of getting heads or tails. However, what if they had a more complex contest? Suppose they flipped coins repeatedly, where Orville would win if two heads appeared in a row, and Wilbur would win if heads was immediately followed by tails. Would each brother still have an equal chance of winning?
Initially, it might seem that they would have the same chance, as there are four combinations for two flips, each with a 25% probability. However, this intuition is incorrect. Wilbur actually has a significant advantage in this scenario.
To understand why, we can visualize the sequence of coin flips as a board game, where each flip determines the path taken. The goal is to reach the finish line. The heads/tails board and the heads/heads board have a critical difference: the heads/heads path has a move that sends you back to the start, which the heads/tails path does not. This is why heads/heads takes longer on average.
Using probability and algebra, we can calculate the average number of flips needed for each combination. For the heads/tails board, let’s define x as the average number of flips to advance one step. There are two options: if tails is flipped, we stay in place and need to flip x more times, resulting in an average of x + 1 flips. If heads is flipped, we advance in one flip. Combining these options gives us an average of two flips to advance one step, leading to four flips to advance two steps.
For the heads/heads board, let y be the average number of flips to finish. There are two options for the first move. If tails is flipped, we return to the start, resulting in an average of y + 1 flips. If heads is flipped, there are two cases: if the next flip is heads, we finish in two flips; if it’s tails, we return to the start and need an average of y + 2 flips. Solving this gives us an average of six flips for heads/heads and four flips for heads/tails.
In practice, this means that heads/heads takes longer to achieve than heads/tails. Ultimately, the Wright brothers only flipped the coin once, and Wilbur won. However, it didn’t matter, as Wilbur’s flight failed, and Orville made aviation history instead.
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This version maintains the essential information while streamlining the content for clarity.
Coin – A flat, typically round piece of metal used as money, often used in probability experiments to demonstrate random outcomes. – In our math class, we used a coin to demonstrate how probability works by predicting the outcome of a coin toss.
Flip – To toss a coin into the air and let it land to show one of its two sides, used in probability to demonstrate random events. – When you flip a coin, there is an equal chance of it landing on heads or tails.
Probability – A measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. – The probability of rolling a three on a standard six-sided die is 1/6.
Chance – The likelihood or possibility of a particular outcome occurring, similar to probability. – There is a 50% chance of getting heads when you flip a fair coin.
Heads – The side of a coin that typically features a portrait or main design, used in probability experiments. – If you flip a coin and it lands on heads, you win the game.
Tails – The opposite side of a coin from heads, often featuring a different design, used in probability experiments. – The coin landed on tails, so we had to try again to get heads.
Average – A number expressing the central or typical value in a set of data, calculated by dividing the sum of the values by their number. – To find the average score of the class, add all the scores together and divide by the number of students.
Outcomes – The possible results of a probability experiment or event. – When rolling a six-sided die, the possible outcomes are 1, 2, 3, 4, 5, and 6.
Algebra – A branch of mathematics dealing with symbols and the rules for manipulating those symbols, used to express mathematical relationships. – In algebra, we often solve equations to find the value of unknown variables.
Game – An activity with rules and objectives, often used in probability to study random events and outcomes. – We played a probability game in class to understand how different outcomes can occur by chance.
