91Ó°ÊÓ

The Binomial Probability Formula is crucial for finding the probability of getting exactly a certain number of successes in a fixed number of independent trials. The formula is: \[P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}\] This formula involves the combination \(\binom{n}{x}\), also known as 'n choose x'. For this exercise, \(n = 40\), \(p = 0.25\), and \(X = 30\).

• Step 1: Calculate the binomial coefficient \(\binom{n}{x}\):\[\binom{40}{30} = \frac{40!}{30!(40-30)!}\]
• Next, compute the probability: \[P(X=30) = \binom{40}{30} (0.25)^{30} (0.75)^{10}\]. This precise calculation tells us the exact likelihood of getting 30 successes in 40 trials with a success chance of 0.25 in every trial.

Breaking this down into smaller steps helps understand how to deal with each part of the formula.
It's always good to check your calculations twice for accuracy.

Normal Approximation is a useful method when dealing with large values of n in binomial distributions. However, certain conditions must be met to apply this approximation accurately. These conditions are:

• \(np \geq 5\)
• \(n(1-p) \geq 5\)

In our exercise, we have:
\[np = 40 \times 0.25 = 10\]
\[n(1-p) = 40 \times 0.75 = 30\]
Both values exceed 5, so the conditions for using the normal approximation are satisfied. Meeting these requirements ensures that the binomial distribution is close enough to the normal distribution for the approximation to be accurate.
This step validates whether we can proceed, saving us from potentially inaccurate approximations if the conditions weren't met.

When approximating a binomial distribution with a normal distribution, a continuity correction must be applied. This adjustment ensures better accuracy by accounting for the fact that the normal distribution is continuous, whereas the binomial is discrete. For our problem, where \(X = 30\), we use \(X = 30.5\) in our approximation. The next step is converting to the standard normal variable \(Z\) using the formula:
\[Z = \frac{X - \text{μ} }{\text{σ}}\] where \( μ = np\) and \(σ = \sqrt{np(1-p)}\). From our previous step, we know:

• \(\text{μ} = 10\)
• \(\text{σ} \approx 2.738\)

Plugging in:
\[Z = \frac{30.5 - 10}{2.738} \approx 7.48\]
Now, using standard normal distribution tables or a calculator, we find that the area to the right of \(Z = 7.48\) is extremely small, indicating \(P(X = 30) \approx 0\).
The continuity correction helps refine our approximation to closely match the exact binomial probability.