Aptitude- Numerical Word Problems - Probability & Combinatorics

In the "Smartest and the Brightest" college there are 180 elite students. 70% of them major in physics, 20% major in math and 5% major in both.
How many students major in unmentioned subjects?

When a client of the "Money & Money" bank wishes to take a loan, his request must go through the proper channels. The request must first be reviewed by the manager and then by the loan consultant, who will review the case independently. Each of them recommends whether to approve or decline the request. Only a request that has been approved by both the manager and the consultant can be implemented.
The likelihood that a request will be approved by the manager is 90%. The likelihood that a request will be approved by the loan consultant is 80%.

What are the odds of being approved for a loan?

When a "Mackelmore & Sons" bank client wishes to take a loan, his request must go through the proper channels. The request must first be reviewed by the manager and then by the loan consultant, who will review the case independently. Each of them recommends whether to approve or decline the request. Only a request that has been approved by both the manager and the consultant can be implemented.

The likelihood that a request will be approved by the manager is 90%.
The likelihood that a request will be approved by the loan consultant 80%.

What are the chances of the decisions made by the manager and the consultant being different?

If two dice are thrown, what is the probability that the sum of the two dice will be a prime number?

In a certain class, 30% of the students like to play basketball, 35% of the students like to play football, and 15% of the students like to play both basketball and football. What is the probability that a randomly selected student either does not like to play neither basketball nor football OR only likes to play basketball?

A bag of 100 blue and purple marbles has three times as many blue marbles as purple marbles. If purple marbles are added to the bag until the number of purple marbles in the bag is double the original amount, what is the probability of taking out a blue marble and then choosing another blue marble?

A loaded die is designed to roll a 'six' 5% more often than a normal die. The odds of rolling any of the other five numbers are even. What is the probability that the loaded die will roll at least 2 'six's in 5 rolls?

In a certain horse race, there are four horses running: Apollo, Brilliant, Champion, and Drill. There is a 10% chance that Apollo will win, a 20% chance that Brilliant will win, a 30% chance that Champion will win, and a 40% chance that Drill will win. Which of the following outcomes has the highest probability?

In a small brewery, one employee seals the bottles while another is in charge of labeling them. The meticulous owner personally checks each bottle before it is released for sale and discards any bottle that is not properly sealed or labeled. If the owner finds one out of every 162 bottles to be not properly sealed, and discards in total 3.7% of the bottles, what is the probability for a bottle to be considered as not properly labeled?

A person rolls a die and then flips a coin the number of times on the die. What is the probability that the person will get heads every time the coin is flipped?