Question

The students in one college have the following rating system for their professors:excellent, good, fair, and bad. In a recent poll of the students, it was found that they believe that 20% of the professors are excellent, 50% are good, 20% are fair, and 10% are bad. Assume that 12 professors are randomly selected from the college.
a. What is the probability that 6 are excellent, 4 are good, 1 is fair, and 1 is bad?
b. What is the probability that 6 are excellent, 4 are good, and 2 are fair?
c. What is the probability that 6 are excellent and 6 are good?
d. What is the probability that 4 are excellent and 3 are good?
e. What is the probability that 4 are bad?
f. What is the probability that none is bad?

Answer 1

Multinomial Formula. Suppose a multinomial experiment consists of n trials, and each trial can result in any of kpossible outcomes: E₁, E₂, . . . , E_k. Suppose, further, that each possible outcome can occur with probabilities p₁, p₂, . . . , p_k. Then, the probability (P) that E₁ occurs n₁ times, E₂ occurs n₂ times, . . . , and E_k occurs n_k times is:

P = [ n! / ( n₁! * n₂! * ... n_k! ) ] * ( p₁^n₁ * p₂^n₂ * . . . * p_k^n_k )

where n = n₁ + n₂ + . . . + n_k.

Let k₁is the number of excellent professors, k₂ is the number of good professors, k₃ is the number of fair professors and k₄ is the number of bad professors.

And let p₁ is the probability that a excellent professor is randomly selected, p₂ , p₃ and p₄ are the probability of randomly selecting good, fair and bad professor respectively.

From the information we have,

p₁ = 0.2

p₂ = 0.5

p₃ = 0.2

p₄ = 0.1

a) Probability that 6 are excellent, 4 are good, 1 is fair, and 1 is bad

n = 12

  

= 0.0022176

b) The probability that 6 are excellent, 4 are good, and 2 are fair.

  

where q = 1 - p₁ + p₂ + p₃

= 0.0022176

c) The probability that 6 are excellent and 6 are good.

  

where q = 1 - p₁ + p₂

= 0.000924

d) The probability that 4 are excellent and 3 are good

  

where q = 1 - p₁ + p₂

= 0.0002245