A Streak Does Not Make a Fair Coin “Due”
Suppose a fair coin lands tails five times in a row. Some people expect heads on the next flip because heads seems “due.” But if the coin flips are independent, the five tails do not change the chance of the next result. Under the fair-coin model, the probability of heads on the next flip is still \(0.5\).
This mistaken belief is called the gambler’s fallacy: thinking that a random outcome is more likely to “balance out” a recent streak, even though the trials are independent. The idea can feel persuasive because a long-run pattern may seem like it ought to show up right away. But long-run relative frequencies, as discussed in The Law of Large Numbers in Simulations, describe what tends to happen across many trials. They do not require a short sequence to correct itself on the next trial.
Two ideas are involved in the fair-coin model. Fair means heads and tails each have probability \(0.5\) on a flip. Independent means the result of one flip does not change the probabilities on another flip. These are different properties: fairness describes the chance of each result, while independence describes how results across trials relate. The claim that a coin is not “due” relies on the independence assumption.
In real life, the physical process might not perfectly match the model. For example, a coin could be bent or a flipping method could consistently favor one side. In this tutorial, we use the specified model of a fair coin with independent flips. Within that model, past results do not adjust the chance of the next result.
Separate the Next Flip from the Whole Sequence
A key habit is to identify exactly which probability a question asks for. “What is the probability of heads on the next flip?” is different from “What is the probability of five tails followed by heads?” The first describes one future result. The second describes a particular sequence of results.
For independent flips, the probability of a specified sequence is found by multiplying the probabilities of its results. For a fair coin, each required result has probability \(1/2\). Thus, a longer specified sequence has a smaller probability than a single next result. That does not mean the next result changes after a streak; it means the whole sequence requires several particular outcomes.
For instance, the probability of six tails in a row is \((1/2)^6=1/64\). The probability of tails on the first five flips and heads on the sixth is also \((1/2)^6=1/64\). Each describes one particular six-flip sequence. Once the first five tails have already occurred, however, the question about the sixth flip alone has probability \(1/2\) for heads. The earlier results are part of the history, not additional requirements for the next flip.
The notation \(P(\text{sixth flip is H}\mid\text{first five flips are T})\) means “the probability that the sixth flip is heads, given that the first five flips were tails.” The vertical bar means “given.” Under the independent fair-coin model, this probability is \(1/2\). The condition after the bar tells us what has already happened; independence tells us it does not change the next-flip probability.
Worked Example: Heads After a Run of Tails
Worked Example: Heads After a Run of Tails
A fair coin is flipped independently. The first seven flips are all tails. What is the probability that the eighth flip is heads?
State: We want the probability of heads on the eighth flip, given that the first seven flips were tails.
Plan: Use the independent fair-coin model. Each flip has probability \(1/2\) of heads, and earlier results do not change that probability.
Do: The result of the eighth flip is one flip, so we do not multiply by the probabilities of the seven tails that have already occurred.
Conclude: Given seven tails in a row, the probability that the eighth flip is heads is \(0.5\). The coin is not more likely to land heads simply because the recent results were tails.
Notice the question’s wording: the seven tails are stated as past results, and only the eighth flip is uncertain. If instead the question asked for the probability of all eight results being tails, we would calculate a particular eight-flip sequence. Read carefully to see whether earlier outcomes are already known or are part of the event whose probability is requested.
Worked Example: Comparing Two Six-Flip Sequences
Worked Example: Comparing Two Six-Flip Sequences
For independent flips of a fair coin, compare the probability of the sequence TTTTTT with the probability of TTTTTH. Then explain why this comparison does not mean heads is due after five tails.
First sequence: TTTTTT specifies six tails. Each required result has probability \(1/2\), so multiply six factors of \(1/2\).
Second sequence: TTTTTH specifies five tails followed by one head. It also specifies six results, each with probability \(1/2\).
Interpretation: Before any of the six flips take place, these two exact sequences have equal probability. If five tails have already occurred, the probability that the sixth flip is heads is \(1/2\), not a value increased by the streak. The probability \(1/64\) for TTTTTH includes the five tails as well as the final head; it is not the probability of the final head alone.
Conclude: Both specified six-flip sequences have probability \(1/64\). After the first five tails are known, heads on the sixth flip still has probability \(1/2\). The distinction is between the probability of a whole sequence and the probability of its next result.
The sequence TTTTTH may look like a correction for the preceding tails, but it is just one possible six-flip sequence. TTTTTT is another. Neither is favored by the fair-coin model. The same reasoning applies if the observed run is longer or shorter: the model does not keep a score and adjust the next flip.
What a Streak Does and Does Not Tell You
A streak can make a complete sequence less likely than a shorter event. For example, six tails in a row is less likely than tails on one specified flip because six results must all be tails. But after observing some of those tails, the probability of a future flip is still determined by the model for that flip.
This distinction also helps with questions about several future flips. After a run of tails, the chance of at least one head in the next three flips is not the chance of heads on just the next flip. The event involves three new flips. Using the complement rule from The Complement Rule and Using the Complement for “At Least One”, find one minus the probability that all three are tails.
Worked Example: At Least One Head After a Streak
A fair coin has just landed tails four times in a row. What is the probability of at least one head in the next three independent flips?
Identify the event: “At least one head” in the next three flips is the complement of “no heads” in those flips. No heads means all three are tails.
Find the complement probability: The probability of three tails in a row is the product of three factors of \(1/2\).
Apply the complement rule: Subtract the probability of no heads from 1.
Conclude: The probability of at least one head in the next three flips is \(7/8\), or \(0.875\). This is larger than \(1/2\) because the event includes three opportunities for heads. It is not larger because the previous four flips were tails.
The value \(0.875\) does not mean heads is guaranteed: there is still probability \(0.125\) that all three new flips are tails. Nor does this calculation show that the coin has to make up for the earlier streak. It answers a different question by considering a different event across three future flips.
Common Mistakes and AP Exam Tips
- Claiming that heads becomes more likely after tails. Under the stated independent fair-coin model, a careful answer says the probability of heads on the next flip remains \(1/2\), regardless of the previous results.
- Confusing an exact sequence with one next outcome. The probability of TTTTTH is \(1/64\), while the probability of heads on the sixth flip after five tails have occurred is \(1/2\). State which event you are calculating.
- Multiplying in outcomes that have already happened. If the question asks only about the next flip given an observed streak, do not include the past flips in the probability calculation. If it asks for the probability of the entire sequence before the flips, include every specified result.
- Assuming a streak changes fairness or independence. A streak may look unusual, but it does not itself show that the next outcome’s probability has changed. Changing the model would require evidence or an explicit assumption that the coin is biased or the flips are dependent.
- Using “long run” to promise an immediate correction. The Law of Large Numbers concerns relative frequencies over many repeated trials. It does not say the next flip, or even the next few flips, must balance earlier results.
- Giving a number without naming the event. A full-credit response identifies whether it concerns one next flip, a specified sequence, or an event across several future flips, and connects the calculation to the fair, independent-flip model.
On an AP response, make the independence assumption explicit when it matters. For example: “Because the flips are independent and the coin is fair, the probability of heads on the next flip is \(0.5\), even after five tails.” If calculating a sequence, show the product of the individual probabilities and state that the result is for the complete specified sequence.
Check Your Understanding
Use the independent fair-coin model. State what event each probability describes.
- After six tails in a row, what is the probability that the next flip is tails? Explain briefly.
- What is the probability of the specified sequence TTHH for four independent fair-coin flips?
- Explain the difference between the probability of TTTTH and the probability of heads on the fifth flip given that the first four flips are tails.
- After three tails, what is the probability of at least one head in the next two flips? Show how you use the complement.
- A friend says, “The next flip has to be heads because tails has happened four times.” Name the error and give a correct response under the fair-coin model.