### Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win 2 for each black ball selected and we lose 2 for each black ball selected and we lose 1 for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with each value?

This problem aims to build our understanding of random events and their predictable outputs. The concepts behind this problem are primarily associated with a probability and probability distribution. We can define probability as a way to indicate the occurrence of an unanticipated event, and the probability can be between zero and one. It estimates the […]

### Suppose that A and B are independent events such that the probability that neither occurs is a and the probability of B is b.

Show that: [ boldsymbol{ P(A) = frac{ 1 – b – a }{ 1 – b } }] Read moreIn how many different orders can five runners finish a race if no ties are allowed?The aim of this question is to develop understanding of some of the basic probability and set theory properties for deriving […]

### At a certain college, 6% of all students come from outside the United States. Incoming students there are assigned at random to freshman dorms, where students live in residential clusters of $40$ freshmen sharing a common lounge area.

How many international students would you expect to find in a typical cluster? With what standard deviation? This question aims to find the expected number of international students in a typical cluster along with their standard deviation. Take into consideration what a random variable is: a collection of numerical values resulting from a random process. […]

### You roll a die. If it comes up a 6 you win 100. If not, you get to roll again. If you get a 6 the second time, you win 50. If not, you lose.

– Develop a probability model for the amount you win. – Find the expected amount you win. Read moreIn how many different orders can five runners finish a race if no ties are allowed?This problem aims to find the probability of getting a particular number, say $6$, by rolling a dice and creating a probability […]

### An urn contains 5 white and 10 black balls. A fair die is rolled and that number of balls is randomly chosen from the urn. What is the probability that all of the balls selected are white? What is the conditional probability that the die landed on 3 if all the balls selected are white?

This question aims to find the joint and conditional probabilities. Probability is a measure of the likelihood that an event will occur. Many events are not able to predicted with absolute certainty. We can only expect the probability of an event, i.e., how likely it is to occur, using it. Probability ranges from 0 to 1, where 0 means […]