Fifty-five percent of the applications received for a particular credit card are accepted.
(a)
Among the next ten applications, what is the probability that all will be rejected? (Round your answer to four decimal places.)
0.0003
 
Correct: Your answer is correct.
(b)
Among the next ten applications, what is the probability that all will be accepted? (Round your answer to four decimal places.)
0.0025
 
Correct: Your answer is correct.
(c)
Among the next ten applications, what is the probability that exactly 5 will be accepted? (Round your answer to four decimal places.)
.2340
 
Correct: Your answer is correct.
(d)
Among the next ten applications, what is the probability that fewer than 4 will be accepted? (Round your answer to four decimal places.)
.1359
 
Incorrect: Your answer is incorrect.

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.

Fifty-five percent of the applications received for a particular credit card are accepted.
(a) Among the next ten applications, what is the probability that all will be rejected? (Round your answer to four decimal places.)
(b) Among the next ten applications, what is the probability that all will be accepted? (Round your answer to four decimal places.)
(c) Among the next ten applications, what is the probability that exactly 5 will be accepted? (Round your answer to four decimal places.)
(d) Among the next ten applications, what is the probability that fewer than 4 will be accepted? (Round your answer to four decimal places.)

Explore how to calculate the probabilities of different scenarios for credit card application acceptance using binomial distribution. This detailed solution covers the calculation of all rejected, all accepted, exactly 5 accepted, and fewer than 4 accepted applications. Suitable for students in Grades 11-12.

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