Calculating Binomial Probabilities for Student Outcomes

Answer:

To solve these problems, we need to use the binomial probability formula and the cumulative distribution function (CDF). The binomial probability formula is given by:

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

where:

( n ) is the number of trials (students in this case),
( k ) is the number of successes (students meeting a certain criterion),
( p ) is the probability of success on an individual trial,
( \binom{n}{k} ) is the binomial coefficient, calculated as ( \frac{n!}{k!(n-k)!} ).

Let's assume you have the values for ( n ) (number of students), ( p ) (probability of success), and ( k ) (number of successes). For the sake of this example, let's say ( n = 26 ) and ( p ) is some probability value (you would need to provide this).

a. Exactly 26 students: ( P(X = 26) )

Using the binomial formula:

[ P(X = 26) = \binom{26}{26} p^{26} (1-p)^{0} = 1 \cdot p^{26} \cdot 1 = p^{26} ]

b. At most 24 students: ( P(X \leq 24) )

This is the cumulative probability for ( X ) from 0 to 24:

[ P(X \leq 24) = \sum_{k=0}^{24} P(X = k) = \sum_{k=0}^{24} \binom{26}{k} p^k (1-p)^{26-k} ]

c. At least 24 students: ( P(X \geq 24) )

This can be calculated as:

[ P(X \geq 24) = 1 - P(X \leq 23) ]

Where:

[ P(X \leq 23) = \sum_{k=0}^{23} P(X = k) = \sum_{k=0}^{23} \binom{26}{k} p^k (1-p)^{26-k} ]

d. Between 20 and 25 students: ( P(20 \leq X \leq 25) )

This is the sum of the probabilities from 20 to 25:

[ P(20 \leq X \leq 25) = \sum_{k=20}^{25} P(X = k) = \sum_{k=20}^{25} \binom{26}{k} p^k (1-p)^{26-k} ]

Summary