Grade 12 · Binomial probability · 578 of 600
Binomial probability · Trial count n=15; Success probability p=0.95
Use success count k=9. Calculate P(X=k), P(X≤k), and the expected count. Givens: Trial count n=15; Success probability p=0.95.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Let X count successes in n=15 independent Bernoulli trials, each with probability p=0.95. Use success count k=9. Calculate P(X=k), P(X≤k), and the expected count.
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A starting point
Use the binomial coefficient for an exact count. For the cumulative probability, add the masses for j=0 through k inclusive.
Work through the reasoning
Step 1
Identify the model and target
Let X count successes in n=15 independent Bernoulli trials, each with probability p=0.95. Use success count k=9. The governing relation is P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ; E[X]=np. Use the binomial coefficient for an exact count. For the cumulative probability, add the masses for j=0 through k inclusive.
Step 2
Substitute and calculate
For k=9, C(15,9)(0.95)^9(1−0.95)^6=0.000049. Adding this formula from k=0 through 9 gives 0.000053; the mean is 15(0.95)=14.25.
Step 3
Check the mathematical meaning
The point mass 0.000049 cannot exceed the cumulative probability 0.000053, and both lie in [0,1]. The expectation np=14.25 need not be an integer even though every observed count is an integer. Success count: 9; Exact mass P(X=k): 0; Cumulative P(X≤k): 0; Mean np: 14.25. Decimal values are rounded, so use unrounded intermediate values.
The answer
For k=9, C(15,9)(0.95)^9(1−0.95)^6=0.000049. Adding this formula from k=0 through 9 gives 0.000053; the mean is 15(0.95)=14.25. The point mass 0.000049 cannot exceed the cumulative probability 0.000053, and both lie in [0,1]. The expectation np=14.25 need not be an integer even though every observed count is an integer. Animation check: Success count: 9; Exact mass P(X=k): 0; Cumulative P(X≤k): 0; Mean np: 14.25. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
Counting successes in a fixed set of independent, identical-probability trials gives a binomial model. Its probability masses belong to integer outcomes, while the expected count may be fractional. Scanning exact masses distinguishes a theoretical distribution from the noisy frequencies of one simulated sample. Let X count successes in n=15 independent Bernoulli trials, each with probability p=0.95. Use success count k=9. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ; E[X]=np
03 · Reflect and transfer
Explain what changes and why.
Explain why a single-count probability and a probability including all smaller counts answer different questions.
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.