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Probability of a Run of Successes

Learn to multiply the probabilities of independent successes to find the chance of a specified consecutive run, and check whether the model fits the situation.

Beginner 8 min read

What You'll Learn

  • Define a run as successes on consecutive attempts.
  • Calculate the probability of a specified run when each attempt has the same success probability.
  • Multiply different success probabilities when consecutive attempts are independent but not identical.
  • Check whether independence and a constant success probability are reasonable.
  • Distinguish a specified run from a run that could occur anywhere in a longer sequence.
  • Communicate a run probability and its assumptions clearly in context.

What Counts as a Run?

In Probability of At Least One in Independent Trials, you found the chance of one or more successes across several attempts by using a complement. A different question focuses on the order of the attempts: what is the probability of succeeding on three attempts in a row?

A run is a sequence of the same outcome on consecutive attempts. For example, making free throws on attempts 2, 3, and 4 is a run of three made shots. The attempts must be next to each other; a miss between two makes breaks up the run. In this tutorial, we will find the probability of success on a specified set of consecutive attempts, such as making attempts 2, 3, and 4.

Definition: A specified run of \(k\) successes occurs when the outcome of interest happens on each of \(k\) identified consecutive attempts. If those attempts are independent and each has success probability \(p\), the probability of that run is \(p^k\).

This follows from the multiplication rule for independent events, introduced in The Multiplication Rule for Independent Events. A run of three successes requires the first success and the next success and the next success. Because the attempts are independent, multiply their probabilities. When each probability is \(p\), multiplying \(p\) by itself \(k\) times gives \(p^k\).

$$ P(\text{success on each of }k\text{ specified consecutive attempts}) =\underbrace{p \cdot p \cdots p}_{k\text{ factors}} =p^k $$

For example, if the chance of success on each attempt is \(0.8\), the chance of success on three specified consecutive attempts is \(0.8^3\), not \(3(0.8)\). Multiplication accounts for the requirement that every attempt in the run succeeds.

Check the Run and the Trial Model

Before calculating, state which attempts make up the run. “Make the next three free throws” identifies three consecutive attempts. “Make three free throws sometime during practice” does not specify their positions and could describe several possible sequences. Those are different probability questions.

The formula \(p^k\) applies when the specified attempts are independent and share the same probability of success. Independence means that the result of one attempt does not change the probability of success on another. A constant \(p\) means that the success probability is the same for every attempt in the run.

Conditions: Use \(p^k\) for a specified run when:
  • The event requires success on each of \(k\) identified consecutive attempts.
  • The attempts are independent.
  • Each attempt has the same probability \(p\) of success.

The setting needs to support these assumptions. For instance, a hypothetical practice model might treat each free throw as independent with a stable success probability. In an actual sequence, fatigue or a change in technique might affect later attempts. If the chance of success changes from one attempt to the next, use the individual probabilities instead of \(p^k\), provided the attempts are still independent.

This is also not the same as finding the chance of at least one run somewhere in a longer sequence. If there are several possible locations for the run, those possibilities can overlap. The formula \(p^k\) gives the probability for the particular consecutive attempts named in the question; by itself, it does not count every possible location.

Calculating a Specified Run

A useful way to organize the calculation is to name the success, identify the consecutive attempts, and then write one probability factor for each required success. Check that every factor refers to the correct attempt. If the probabilities are equal, the product can be written as a power.

Formula: If the \(k\) specified consecutive attempts are independent and each has probability \(p\) of success, then $$ P(\text{specified run of }k\text{ successes})=p^k. $$ If the independent attempts have different success probabilities \(p_1,p_2,\ldots,p_k\), then $$ P(\text{success on all }k\text{ specified attempts})=p_1p_2\cdots p_k. $$

The second formula is still the multiplication rule for independent events. The equal-probability case is a convenient special case: if every \(p_i=p\), the product \(p_1p_2\cdots p_k\) becomes \(p^k\). Neither formula applies automatically just because attempts happen in a sequence; the independence assumption matters.

Worked Examples

Worked Example: Three Free Throws in a Row

In a hypothetical practice model, a basketball player makes a free throw with probability \(0.80\) on each attempt. Assume the attempts are independent and the success probability stays the same. Find the probability that the player makes the next three free throws.

State. The event is making all three specified consecutive free throws.

Plan. Each attempt has success probability \(p=0.80\), and there are \(k=3\) attempts. The stated model gives independent attempts with the same probability, so multiply the three success probabilities.

Do.

$$ P(\text{three makes in a row}) =0.80 \cdot 0.80 \cdot 0.80 =(0.80)^3 =0.512 $$

Check the product by first calculating \(0.80^2=0.64\), then \(0.64(0.80)=0.512\). Thus, the probability rounded to four decimal places is \(0.5120\).

Conclude. Under the stated assumptions, the probability that the player makes the next three free throws is \(0.512\), or \(51.2\%\).

Worked Example: Four Consecutive Saves

In a hypothetical training drill, a goalie has a \(0.70\) probability of saving each shot faced. Suppose the outcomes of the shots are independent and the goalie’s save probability stays constant. What is the probability that the goalie saves the next four shots in a row?

State. We want the probability of saves on all four specified consecutive shots.

Plan. A save is the success, with \(p=0.70\), and the run has \(k=4\) shots. Since the attempts are modeled as independent with a common success probability, use \(p^k\).

Do.

$$ P(\text{four saves in a row}) =(0.70)^4 =0.70 \cdot 0.70 \cdot 0.70 \cdot 0.70 =0.2401 $$

As a check, \(0.70^2=0.49\), and \(0.49^2=0.2401\). The probability rounded to four decimal places is \(0.2401\).

Conclude. Under the training model, the probability that the goalie saves the next four shots is \(0.2401\), or \(24.01\%\).

Worked Example: Independent Attempts with Different Probabilities

A hypothetical sensor runs three checks in a row. The probabilities of correctly detecting a signal on checks 1, 2, and 3 are \(0.90\), \(0.80\), and \(0.75\), respectively. Assume the check outcomes are independent. Find the probability that all three checks detect the signal.

State. The event is a successful detection on each of the three specified consecutive checks.

Plan. The checks are independent, but their success probabilities are not all the same. Use the general multiplication rule for independent events and multiply the three different probabilities.

Do.

$$ P(\text{three detections}) =0.90 \cdot 0.80 \cdot 0.75 =0.72 \cdot 0.75 =0.54 $$

Check by multiplying in another order: \(0.80 \cdot 0.75=0.60\), and \(0.90 \cdot 0.60=0.54\). The probability rounded to four decimal places is \(0.5400\).

Conclude. Under the stated independence model, the probability that all three checks detect the signal is \(0.54\), or \(54\%\).

Common Mistakes and AP Exam Tips

  • Adding the success probabilities. A run requires success on every specified attempt, so use the multiplication rule when the attempts are independent. Adding probabilities does not represent “success on all of them.”
  • Using \(p^k\) when the probabilities differ. If independent attempts have different success probabilities, multiply those individual probabilities, as in the sensor example. Use \(p^k\) only when the probability is the same on each attempt.
  • Forgetting what “consecutive” means. A miss between two successes breaks the run. Identify the exact adjacent attempts the question is asking about.
  • Assuming any run and a specified run are the same event. \(p^k\) finds the probability of success on one named block of \(k\) attempts. A run that could appear in different positions in a longer sequence is a different event.
  • Claiming independence without a reason. State that the model treats attempts as independent, and consider whether one result could change the chance on the next attempt. Repeated attempts are not automatically independent.
  • Reporting a number without interpreting it. Include the event, the assumptions, and the probability in context. Keep full precision during multiplication and round the final result consistently.
AP Exam Tip: For a complete response, define success, identify the specified consecutive attempts, justify independence and equal probabilities when using \(p^k\), show the multiplication, and interpret the result in context. If the probabilities differ, write each factor rather than forcing the calculation into a power.

Key Takeaway

A specified run requires success on every attempt in a consecutive block. For independent attempts, multiply the success probabilities. If the probability is the same on all \(k\) attempts, the product is \(p^k\); if it differs, use each attempt’s probability as its own factor.

Key takeaway: The probability of success on \(k\) specified consecutive independent attempts is \(p^k\) when each attempt has the same success probability \(p\). Be precise about which attempts define the run, and check the independence and probability assumptions.

Check Your Understanding

For each question, identify the required consecutive successes, state the independence assumption, and show the calculation.

  1. A hypothetical bowler has a \(0.60\) probability of knocking down all the pins on each frame. Assuming independent frames with the same probability, find the probability of strikes on three specified consecutive frames.
  2. A sensor has a \(0.85\) probability of detecting a signal on each independent check. Find the probability of detection on five specified consecutive checks.
  3. Three independent checks have success probabilities \(0.50\), \(0.80\), and \(0.90\). Find the probability that all three are successful.
  4. Explain why a failure between two successes means the successes do not form one consecutive run.
  5. In your own words, explain why \(p^k\) does not by itself find the probability of a run that could occur anywhere in a longer sequence.

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