This problem involves determining the probability that more than 20% of the students in Mr. C's class have the flu next week. Here's how you can approach solving it and how to use the TI calculator (TI-83, TI-84, etc.):

Key Information:

• Probability of a student having the flu ($p$): 0.3
• Number of students ($n$): 25
• If more than 20% of the students have the flu, the test will be canceled. $20\%$ of 25 students is $0.20 \times 25 = 5$ students.
• So, we are looking for the probability that more than 5 students have the flu. This means we want to find $P(X > 5)$, where $X$ is the number of students with the flu.

Step-by-Step Process:

1. Identify the type of distribution:

   • This is a binomial probability distribution because we are dealing with a fixed number of students (25), each with the same probability (0.3) of having the flu.
   • The question asks for the probability of "more than 5" students having the flu, so you will use binomial cumulative distribution.

2. Set up the binomial problem:

   • The random variable $X$ represents the number of students who have the flu.
   • $X \sim Binomial(n = 25, p = 0.3)$

3. Complement Rule:

   • Instead of calculating $P(X > 5)$ directly, we can use the complement rule. The complement is $P(X \leq 5)$, so: $P(X > 5) = 1 - P(X \leq 5)$
   • This simplifies the calculation.

Instructions for TI Calculator (TI-84 or similar):

1. Use the binomcdf function:

   • Since you're finding the cumulative probability $P(X \leq 5)$, you should use the binomcdf function (cumulative distribution function).

2. Steps to enter on TI calculator:

   1. Press the 2nd button.
   2. Press VARS to access the distribution menu.
   3. Scroll down to binomcdf( (option A on some models).
   4. Input the parameters: $n = 25$, $p = 0.3$, and $x = 5$ (since we want $P(X \leq 5)$).
      • It should look like this: binomcdf(25, 0.3, 5)
   5. Press Enter.
   6. The calculator will give you the cumulative probability $P(X \leq 5)$.

3. Calculate $P(X > 5)$:

   • Once you have the value for $P(X \leq 5)$, subtract it from 1: $P(X > 5) = 1 - P(X \leq 5)$

4. Interpret the result:

   • The result will give you the probability that more than 20% of the students (i.e., more than 5 students) will have the flu, which is the probability that the test will be canceled.

Which to Use: binompdf or binomcdf?

• Use binomcdf because you are interested in cumulative probabilities (i.e., the probability that $X$ is less than or equal to a certain number). The binompdf function only gives the probability for an exact number of successes (in this case, an exact number of students with the flu), but we need the probability for "less than or equal to" 5 students, so binomcdf is the right choice.

Additional Help?

Would you like more detailed steps on interpreting the calculator's output or further explanation of the complement rule?

---

Relative Questions:

1. What is the exact probability that exactly 5 students will have the flu?
2. What if the probability of flu increased to 0.4? How would that affect the test cancellation probability?
3. How can you apply this method to a class size of 30 students?
4. What is the difference between binompdf and binomcdf in other contexts?
5. How would this change if the threshold was 30% of students instead of 20%?

Tip:

Always check if you're asked for "more than" or "less than" probabilities to determine whether you need the complement rule.