Questions: Use the binomial probability distribution to answer the following questions. Researchers conducted a study to determine whether there were significant differences in graduation rates between medical students admitted through special programs (such as affirmative action) and medical students admitted through the regular admissions criteria. It was found that the graduation rate was 72% for the medical students admitted through special programs. (a) If 13 of the students from the special programs are randomly selected, find the probability that exactly 10 of them graduated. (b) If 13 of the students from the special programs are randomly selected, find the probability that fewer than 7 of them graduated. (c) If 13 of the students from the special programs are randomly selected, find the probability that more than 5 of them graduated.

Use the binomial probability distribution to answer the following questions. Researchers conducted a study to determine whether there were significant differences in graduation rates between medical students admitted through special programs (such as affirmative action) and medical students admitted through the regular admissions criteria. It was found that the graduation rate was 72% for the medical students admitted through special programs. (a) If 13 of the students from the special programs are randomly selected, find the probability that exactly 10 of them graduated. (b) If 13 of the students from the special programs are randomly selected, find the probability that fewer than 7 of them graduated. (c) If 13 of the students from the special programs are randomly selected, find the probability that more than 5 of them graduated.
Transcript text: Use the binomial probability distribution to answer the following questions. Researchers conducted a study to determine whether there were significant differences in graduation rates between medical students admitted through special programs (such as affirmative action) and medical students admitted through the regular admissions criteria. It was found that the graduation rate was $72 \%$ for the medical students admitted through special programs. (a) If 13 of the students from the special programs are randomly selected, find the probability that exactly 10 of them graduated. (b) If 13 of the students from the special programs are randomly selected, find the probability that fewer than 7 of them graduated. (c) If 13 of the students from the special programs are randomly selected, find the probability that more than 5 of them graduated.
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Solution

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Solution Steps

Step 1: Probability of Exactly 10 Graduates

To find the probability that exactly 10 out of 13 students graduated, we use the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

  • \( n = 13 \) (number of trials),
  • \( x = 10 \) (number of successes),
  • \( p = 0.72 \) (probability of success),
  • \( q = 1 - p = 0.28 \) (probability of failure).

Calculating this gives:

\[ P(X = 10) = 0.2351 \]

Step 2: Probability of Fewer Than 7 Graduates

To find the probability that fewer than 7 students graduated, we sum the probabilities from 0 to 6:

\[ P(X < 7) = \sum_{x=0}^{6} P(X = x) \]

The calculated probability for this scenario is:

\[ P(X < 7) = 0.0440 \]

Step 3: Probability of More Than 5 Graduates

To find the probability that more than 5 students graduated, we sum the probabilities from 6 to 13:

\[ P(X > 5) = \sum_{x=6}^{13} P(X = x) \]

The calculated probability for this scenario is:

\[ P(X > 5) = 0.9882 \]

Final Answer

  • Probability that exactly 10 students graduated: \( \boxed{0.2351} \)
  • Probability that fewer than 7 students graduated: \( \boxed{0.0440} \)
  • Probability that more than 5 students graduated: \( \boxed{0.9882} \)
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