What is the definition of probability?

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. An event with a probability of 0 will not occur, while an event with a probability of 1 will occur.

How do you calculate the probability of a single event occurring?

The probability of a single event occurring can be calculated using the formula: 
P(A) = Number of favorable outcomes / Total number of possible outcomes. 
For Example, the probability of rolling a 3 on a fair six-sided die is P(3) = 1/6.

What is the difference between independent and dependent events?

Independent events are those whose outcomes do not affect each other. For example, flipping a coin and rolling a die are independent. Dependent events are those where the outcome of one event affects the outcome of another. For example, drawing cards from a deck without replacement.

If you roll two six-sided dice, what is the probability that the sum of the numbers rolled is 7?

To find the probability of rolling a sum of 7, identify the favorable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — a total of 6 outcomes. 
The total number of outcomes when rolling two dice is 
6 x 6 = 36. 
Thus, 
P(sum = 7) = 6/36 
= 1/6.

Define the mean, median, and mode in a data set.

The mean is the average of all data points, calculated by summing them and dividing by the number of points. The median is the middle value when the data points are arranged in order. The mode is the value that appears most frequently in the data set.

What is the formula for the expected value (EV) of a discrete random variable?

The expected value is calculated using the formula: 
EV = Σ [xP(x)], where x represents the value of each outcome and P(x) is the probability of that outcome.

What is the formula for the variance of a data set?

Variance is calculated using the formula: 
Var(X) 
= Σ [(x - μ)² * P(x)], where μ is the mean of the data set.

What is a common mistake when calculating probabilities?

A common mistake is to forget to adjust the total number of outcomes when dealing with dependent events. Always ensure you are using the correct total based on whether the events are independent or dependent.

A bag contains 3 red balls and 2 blue balls. If one ball is drawn at random, what is the probability that it is red?

The total number of balls is 5. The number of favorable outcomes (red balls) is 3. Thus, P(red) = 3/5.

If a fair coin is flipped three times, what is the probability of getting exactly two heads?

The total outcomes when flipping a coin three times is 2³ = 8. 
The favorable outcomes for exactly two heads (HHT, HTH, THH) are 3. Thus, P(exactly 2 heads) 
= 3/8.

In a normal distribution, what percentage of data falls within one standard deviation of the mean?

Approximately 68% of the data falls within one standard deviation of the mean in a normal distribution.

In a dataset, the numbers are 10, 15, 20, 25, and 30. What is the median?

To find the median, arrange the numbers in order. The median is the middle value, which is 20, since there are 5 data points.

What is the relationship between combinations and permutations?

Combinations are used when the order does not matter, calculated as C(n, k) = n! / (k! * (n - k)!), while permutations are used when the order matters, calculated as P(n, k) = n! / (n - k)!. 
For Example, choosing 2 out of 3 items can be done in 3 ways (combinations) or 6 ways (permutations).

A die is rolled twice. What is the probability that the sum of the numbers rolled is greater than 8?

First, calculate the total outcomes: 6 * 6 = 36. The favorable outcomes for sums greater than 8 are: (3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6) — totaling 10 outcomes. Thus, P(sum > 8) = 10/36 = 5/18.

How do you calculate the median of an even set of numbers?

To calculate the median of an even set, arrange the numbers in order and take the average of the two middle numbers.