The random variable X is such that X ~ B(n, p)
The mean value of X is 225
The variance of X is 144
Find p - AQA - A-Level Maths: Mechanics - Question 11 - 2021 - Paper 3
Question 11
The random variable X is such that X ~ B(n, p)
The mean value of X is 225
The variance of X is 144
Find p.
Circle your answer.
Worked Solution & Example Answer:The random variable X is such that X ~ B(n, p)
The mean value of X is 225
The variance of X is 144
Find p - AQA - A-Level Maths: Mechanics - Question 11 - 2021 - Paper 3
Step 1
Find p using the mean value
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Answer
In a binomial distribution, the mean (μ) is given by the formula:
extMean=nimesp
From the question, we know that the mean value is 225. Therefore:
225=nimesp
Hence, we can express p as:
p=n225
Step 2
Find p using the variance
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Answer
The variance (σ²) in a binomial distribution is calculated using the formula:
extVariance=nimespimes(1−p)
Given that the variance is 144, we can write:
144=nimespimes(1−p)
Substituting the value of p:
144=n×(n225)×(1−n225)
This simplifies to:
144=225×(1−n225)
Step 3
Solve for n and then for p
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Answer
Rearranging gives:
1−n225=225144
Thus:
n225=1−225144
This leads to:
n225=22581
From here, we can solve for n:
n=81225×225=675
Now substituting n back into the equation for p:
p=675225=0.33
Step 4
Select the closest value
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Answer
Since 0.33 is approximately equal to 0.36, we can circle the best matching answer:
