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Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Identify a valid binomial count model.
- Use the complement of at least one.
- Include the number of possible arrangements for an exact count.
Before you start
Powers, probabilities of independent events, and basic combinations.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Five independent sensor checks each have a false-alarm probability of 0.1. What is the probability of at least one false alarm, and what is the probability of exactly two?
Why this math matters
Repeating a low-probability event creates more opportunities for it to occur. A binomial model counts successes across a fixed number of comparable independent trials. Here success is simply the event being counted, a false alarm; it is not a judgment that the event is desirable. Checking the conditions comes before choosing the formula.
Set up the model
A useful answer starts with clear assumptions:
- There are exactly five checks, regardless of their results.
- Each check has the same false-alarm probability 0.1.
- Check outcomes are independent; shared interference would require a different model.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
How often will a small batch include a false alarm?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Five independent sensor checks each have a false-alarm probability of 0.1. What is the probability of at least one false alarm, and what is the probability of exactly two?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Define the count
X = number of false alarms; X ~ Binomial(5, 0.1)
The possible count values run from zero through five. Fixed trial count, common probability, and independence justify this model.
Subtract the no-alarm case
P(X ≥ 1) = 1 − P(X = 0) = 1 − 0.9⁵ = 0.40951
All five checks must avoid a false alarm for the complement to occur. Multiplication is justified by independence.
Count exactly two alarms
P(X = 2) = C(5, 2)(0.1)²(0.9)³ = 10(0.01)(0.729) = 0.0729
There are ten possible pairs of alarming checks. Each specified arrangement has the same probability, so their probabilities add.
The result
At least one false alarm has probability 40.951%; exactly two has probability 7.29%.
Expected count is np = 0.5 alarms, but that is not the probability of any particular event. The at-least-one probability is already about 41% despite each individual check having only a 10% probability.
Common mistakes to catch
- Multiplying 5 by 0.1 gives an expected count, not the exact at-least-one probability.
- Omitting the combination factor counts only one arrangement of the two alarms.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
For 20 independent checks with p = 0.05, what is the expected false-alarm count?
Show a hint
Use np.
Reveal answer and explanation
1
20(0.05) = 1 expected alarm, although an actual count may be zero or greater than one.
Practice 2
Under that same 20-check model, what is the probability of no false alarms?
Show a hint
All checks must avoid an alarm.
Reveal answer and explanation
0.95²⁰ ≈ 0.3585
Multiply the independent no-alarm probability 0.95 twenty times.
Take the idea with you
Before using a binomial formula for surveys, inspections, or trials, explain why its independence and common-probability assumptions fit.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: What does two standard deviations mean for package weights?
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