In a university, 30% of the students major in Business management, 25% major in mathematics and 10% major in both business management and mathematics. A student from this university is selected at random.

A) what is the probability that the student majors in business management or mathematics?
B) what is the probability that the student majors in neither of these two courses?

A = Business management

B = mathematics

a)
P(A U B) = P(A) + P(B) - P(A ∩ B)
= 0.3 + 0.25 - 0.1
= 0.45

b)
P(not A and not B)
= P(A' ∩ B')
= 1 - P(A U B)
= 1 - 0.45
= 0.55

Suppose that a drug is known to be 90% effective in treating a certain disease. What is the probability that it will be successful in treating seven to ten out of 12 patients with the disease?

A) To find the probability that the student majors in business management or mathematics, we need to add the probabilities of these two events, and then subtract the probability of majoring in both.

The probability of majoring in business management is 30% or 0.30.
The probability of majoring in mathematics is 25% or 0.25.
The probability of majoring in both business management and mathematics is 10% or 0.10.

So, the probability that the student majors in business management or mathematics is:

P(Business or Math) = P(Business) + P(Math) - P(Business and Math)
= 0.30 + 0.25 - 0.10
= 0.45 or 45%

Therefore, the probability that the student majors in business management or mathematics is 45%.

B) To find the probability that the student majors in neither business management nor mathematics, we need to subtract the probability of majoring in business management or mathematics from 100%.

Since we already know that the probability of majoring in business management or mathematics is 45%, the probability of not majoring in either is:

P(Neither) = 1 - P(Business or Math)
= 1 - 0.45
= 0.55 or 55%

Therefore, the probability that the student majors in neither business management nor mathematics is 55%.

To calculate the probability, we need to understand the concepts of union and intersection. The union of two sets represents the combination of elements from both sets, while the intersection represents the common elements between the sets.

Let's break down the problem and calculate the probabilities:

A) To find the probability that the student majors in Business Management or Mathematics, we need to calculate the union of these two sets.

1. The probability of majoring in Business Management is 30%.
2. The probability of majoring in Mathematics is 25%.
3. The probability of majoring in both Business Management and Mathematics is 10%.

To calculate the probability of the union, we can use the formula:
P(A or B) = P(A) + P(B) - P(A and B)

Thus, the probability that the student majors in Business Management or Mathematics is calculated as follows:
P(Business Management or Mathematics) = P(Business Management) + P(Mathematics) - P(Business Management and Mathematics)
= 30% + 25% - 10%
= 45%.

Therefore, the probability that the student majors in Business Management or Mathematics is 45%.

B) To find the probability that the student majors in neither Business Management nor Mathematics, we need to find the complement of the union of these two sets.

The complement of an event represents all the outcomes that are not part of that event.

The probability of not majoring in Business Management is 100% - 30% = 70%.
The probability of not majoring in Mathematics is 100% - 25% = 75%.

To calculate the probability of neither major, we multiply the probabilities of not majoring in each field:

P(Neither Business Management nor Mathematics) = P(Not Business Management) * P(Not Mathematics)
= 70% * 75%
= 52.5%.

Therefore, the probability that the student majors in neither Business Management nor Mathematics is 52.5%.