To solve this, we're working with a binomial probability distribution. The formula for binomial probability is:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Where:

• $n$ is the number of trials (in this case, 10 applications),
• $k$ is the number of successes (accepted applications),
• $p$ is the probability of success on a single trial (55%, or 0.55),
• $\binom{n}{k}$ is the number of ways to choose $k$ successes from $n$ trials.

Part (d) Calculation: Probability of Fewer than 4 Applications Being Accepted

This means we need to calculate the cumulative probability for $X = 0, 1, 2, 3$. We sum the probabilities for these cases:

$P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

Let me calculate this for you.The probability that fewer than 4 of the next 10 applications will be accepted is approximately $0.1020$ (rounded to four decimal places). This suggests that the original answer of $0.1359$ was incorrect.

Would you like further details on the calculation, or do you have any other questions?

Here are 5 related questions to explore:

1. How does the binomial distribution apply to real-world scenarios?
2. What is the probability of accepting between 4 and 6 applications?
3. How does changing the acceptance rate to 60% affect the results?
4. How would the probability change if we considered 15 applications instead of 10?
5. What is the expected number of accepted applications in this scenario?

Tip: When using the binomial formula, ensure you are consistent with probability values and round your results only after performing all calculations.