Question 1180787: The probability of occurrence of a certain event is equal to 0,6 in each of independent trials. 1000 trials have been made. Find the probability that the relative frequency of occurrence of the event deviates from its probability less than on 0,05.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this probability problem using the normal approximation to the binomial distribution:
**1. Define the parameters:**
* p = 0.6 (probability of the event in a single trial)
* n = 1000 (number of trials)
* ε = 0.05 (maximum allowed deviation of the relative frequency from the probability)
**2. Calculate the mean and standard deviation of the number of successes:**
Since we have a large number of independent trials, we can use the normal approximation to the binomial distribution.
* Mean (μ) = n * p = 1000 * 0.6 = 600
* Standard deviation (σ) = sqrt(n * p * (1 - p)) = sqrt(1000 * 0.6 * 0.4) = sqrt(240) ≈ 15.49
**3. Convert the problem to a standard normal distribution problem:**
We want to find the probability that the relative frequency (which is the number of successes divided by the number of trials) deviates from p by less than ε. This can be written as:
P(|(number of successes / n) - p| < ε)
Or, in terms of the number of successes (x):
P(|x/n - p| < ε)
Multiplying by n:
P(|x - np| < nε)
Substituting our values:
P(|x - 600| < 1000 * 0.05)
P(|x - 600| < 50)
This is equivalent to:
P(550 < x < 650)
Now, we convert to z-scores:
z_lower = (550 - 600) / 15.49 ≈ -3.23
z_upper = (650 - 600) / 15.49 ≈ 3.23
**4. Find the probability using the standard normal distribution table or calculator:**
We want to find P(-3.23 < z < 3.23). This is equal to:
P(z < 3.23) - P(z < -3.23)
Using a standard normal table or calculator:
P(z < 3.23) ≈ 0.9994
P(z < -3.23) ≈ 0.0006
Therefore, P(-3.23 < z < 3.23) ≈ 0.9994 - 0.0006 ≈ 0.9988
**Answer:**
The probability that the relative frequency of occurrence of the event deviates from its probability by less than 0.05 is approximately 0.9988.
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