Of Those Who Take A Certification Exam Pass It Assume Binomi

Question

80 of those who take a certification exam pass it assume binomial Analyze the probability of certain outcomes related to certification exam registration and passing, using the binomial and Poisson distributions. The problem involves calculating the probability that exactly 12 out of 15 registered candidates will pass the exam, assuming an 80% pass rate, and the probability that at least 25 new registrations will occur over the next five days, given an average of 3 registrations per day. The goal is to determine the likelihood of these events based on the specified distributions and parameters.

Dr. Jack HW Helper · Accepted Answer

The analysis of certification exam outcomes involves understanding binomial and Poisson probability distributions. These statistical tools are essential for making predictions in scenarios involving discrete probabilities of specific events over period or sample sizes. Probability that exactly 12 out of 15 candidates pass the exam The binomial distribution is well-suited for modeling the chance of a fixed number of successes in a sequence of independent Bernoulli trials. In this case, each candidate's pass/fail outcome is independent, with a success probability p = 0.8 (or 80%). The total number of candidates, n, is 15. The probability of exactly k successes (passes) is given by: $$ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} $$ Plugging in the values n = 15, k = 12, p = 0.8: $$ P(X = 12) = \binom{15}{12} (0.8)^{12} (0.2)^3 $$ Calculating the binomial coefficient: $$ \binom{15}{12} = \frac{15!}{12! \times 3!} = 455 $$ Calculating the probabilities: $$ (0.8)^{12} \approx 0.0687 $$ $$ (0.2)^3 = 0.008 $$ Therefore: $$ P(X=12) \approx 455 \times 0.0687 \times 0.008 \approx 455 \times 0.0005496 \approx 0.2501 $$ This matches the option 0.2501, confirming that the probability that exactly 12 of the 15 candidates will pass the exam is approximately 0.2501. Probability that at least 25 people will register over 5 days The Poisson distribution models the number of events happening over a fixed period when these events occur independently, with a known average rate λ. Given an average registration rate of 3 people per day, over 5 days, the expected total, λ, is: $$ \lambda = 3 \times 5 = 15 $$ We are asked to find the probability that 25 or more people register in this period: $$ P(X \geq 25) = 1 - P(X \leq 24) $$ Calculating the cumulative probability $ P(X \leq 24) $ directly for Poisson distribution can be done via tables, software, or approximation methods. Using a normal approximation to the Poisson distribution: \[ \text{mean} = \lambda = 15,\quad \text{standard devia

Of Those Who Take A Certification Exam Pass It Assume Binomi

80 of those who take a certification exam pass it assume binomial

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Probability that exactly 12 out of 15 candidates pass the exam

Probability that at least 25 people will register over 5 days

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