Question

a university found that 40% of its students withdraw without completing the introductory statistics course. assume that 20 students registered for the course. a. compute the probability that 2 or fewer will withdraw (to 4 decimals). b. compute the probability that exactly 4 will withdraw (to 4 decimals). c. compute the probability that more than 3 will withdraw (to 4 decimals). d. compute the expected number of withdrawals.

Explanation:

Step1: Identify the distribution

This is a binomial distribution problem. Let \(n = 20\) (number of students) and \(p=0.4\) (probability of withdrawal). The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\).

Step2: Calculate part a

We need \(P(X\leq2)=P(X = 0)+P(X = 1)+P(X = 2)\)
For \(k = 0\):

$$
LATEXBLOCK0
$$

For \(k = 1\):

$$
LATEXBLOCK1
$$

For \(k = 2\):

$$
LATEXBLOCK2
$$

\(P(X\leq2)=0.00003656 + 0.00030467+0.00137102\approx0.0017\)

Step3: Calculate part b

For \(k = 4\):

$$
LATEXBLOCK3
$$

Step4: Calculate part c

\(P(X>3)=1 - P(X\leq3)=1-(P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3))\)
For \(k = 3\):

$$
LATEXBLOCK4
$$

\(P(X\leq3)=0.00003656+ 0.00030467+0.00137102 + 0.0350\approx0.0367\)
\(P(X>3)=1 - 0.0367 = 0.9633\)

Step5: Calculate part d

The expected value of a binomial distribution is \(E(X)=n\times p\). Here, \(n = 20\) and \(p = 0.4\), so \(E(X)=20\times0.4 = 8\)

Answer:

a. 0.0017
b. 0.1244
c. 0.9633
d. 8