The probability a company will hire a new employee is 0.38. Suppose three companies are randomly

selected.

(a) Find the probability all three companies will hire a new employee.

(b) Find the probability at least one company will hire a new employee.

(c) Find the probability none of the three companies will hire a new employee.

binomial distribution

p(n,k) = C(n,k)p^k (1-p)^(n-k)

p(3,1) = C(3,3) .38^3 * .62^0
= 1 * .38^3 * 1

do c next because b is 1 - c

p(3,0) = 1 * 1 * .68^3

now b = 1 - c
= 1 - .68^3

To solve these probability questions, we'll use the concept of independent events and the multiplication rule.

(a) To find the probability that all three companies will hire a new employee, we need to multiply the probabilities of each event happening. Since these events are independent, the probability of all three events occurring is the product of their individual probabilities.

P(all three companies hire) = P(company 1 hires) * P(company 2 hires) * P(company 3 hires)

P(all three companies hire) = 0.38 * 0.38 * 0.38 = 0.0549 (rounded to four decimal places)

So, the probability that all three companies will hire a new employee is approximately 0.0549.

(b) To find the probability that at least one company will hire a new employee, we can use the complement rule. The complement of "at least one company hiring" is "none of the companies hiring." So, we can find the probability of none of the companies hiring and subtract it from 1.

P(at least one company hires) = 1 - P(none of the companies hire)

To find the probability that none of the companies will hire a new employee, we need to find the probability that each individual company does not hire a new employee, and then multiply these probabilities together.

P(none of the companies hire) = P(company 1 does not hire) * P(company 2 does not hire) * P(company 3 does not hire)

P(none of the companies hire) = (1 - P(company 1 hires)) * (1 - P(company 2 hires)) * (1 - P(company 3 hires))

P(none of the companies hire) = (1 - 0.38) * (1 - 0.38) * (1 - 0.38) = 0.2177 (rounded to four decimal places)

Now, we can find the probability that at least one company will hire a new employee:

P(at least one company hires) = 1 - 0.2177 = 0.7823 (rounded to four decimal places)

So, the probability that at least one company will hire a new employee is approximately 0.7823.

(c) We have already calculated the probability that none of the three companies will hire a new employee in part (b). The probability of none of the companies hiring is 0.2177 (rounded to four decimal places).

So, the probability that none of the three companies will hire a new employee is approximately 0.2177.

To find the probability of independent events occurring, we multiply the probabilities of each event. For (a), we multiply the probability that each of the three companies will hire a new employee:

(a) The probability that all three companies will hire a new employee is given by:

P(all three companies) = P(company 1) * P(company 2) * P(company 3)
= 0.38 * 0.38 * 0.38
= 0.0549 (rounded to four decimal places)

So, the probability that all three companies will hire a new employee is approximately 0.0549.

For (b), we need to find the probability that at least one company will hire a new employee. To do this, we can find the probability of the opposite event (i.e., the probability that none of the three companies will hire a new employee) and subtract it from 1:

(b) The probability of at least one company hiring a new employee is given by:

P(at least one company) = 1 - P(none of the three companies)
= 1 - P(company 1)^3
= 1 - (1 - 0.38)^3
= 1 - 0.5325
= 0.4675 (rounded to four decimal places)

So, the probability that at least one company will hire a new employee is approximately 0.4675.

For (c), we need to find the probability that none of the three companies will hire a new employee. This is the opposite event of (b):

(c) The probability that none of the three companies will hire a new employee is given by:

P(none of the three companies) = P(company 1)^3
= (1 - 0.38)^3
= 0.5325 (rounded to four decimal places)

So, the probability that none of the three companies will hire a new employee is approximately 0.5325.