Binomial Distribution

Concept:
Probability Distribution - Binomial Events with Replacement.

Step 1: Identify the random variable and its possible values.
Let denote the number of queens drawn in two successive draws. Since we are drawing two cards, the possible values for are 0, 1, or 2 queens.

Step 2: Calculate the probability of success and failure for a single draw.
In a standard deck of 52 cards, there are 4 queens. Therefore, the probability of drawing a queen is . The probability of drawing a non-queen is .

Step 3: Calculate the probability of drawing exactly 0 queens.
For , neither the first nor the second card is a queen. Since the cards are drawn with replacement, the draws are independent. .

Step 4: Calculate the probability of drawing exactly 1 queen.
For , we can either draw a queen first and a non-queen second, OR a non-queen first and a queen second. .

Step 5: Calculate the probability of drawing exactly 2 queens and compile the distribution.
For , both the first and second draws must yield a queen. The probability is calculated as .
Finally, organizing these calculated probabilities into a table format ( mapping to ) perfectly matches the table provided in option A. $ $

Concept:
Binomial Probability Distribution.

Step 1: Define the binomial distribution parameters.
Let be the total number of problems. Let represent the random variable for the number of problems the candidate is unable to solve.

Step 2: Determine the probability of success and failure.
The probability of solving a problem is . The probability of being unable to solve a problem (which we define as our "success" for variable ) is .

Step 3: Set up the mathematical condition based on the question.
We need the probability that he is unable to solve "less than two" problems. This means , which translates to finding the sum of probabilities for and . Thus, we calculate .

Step 4: Apply the Binomial Probability formula.
Using the formula , we substitute our values:
.

Step 5: Simplify the algebraic expression to find the final result.
First, evaluate the combinations: and . Substituting these back gives .
We can simplify to .
This leaves us with .
Next, factor out the common term from both components to get .
Finally, solve the fraction inside the parentheses: .
This results in the final formatted answer: . $ $

Step 1: Use the formula for the volume of a tetrahedron.
The volume of a tetrahedron with vertices at , , , and is given by: Substitute the coordinates of points into this determinant formula.
Step 2: Compute the determinant.
The determinant calculation gives the volume as cubic units.
Step 3: Conclusion.
Thus, the volume of the tetrahedron is cu. units.

Step 1: Binomial distribution formula.
The probability mass function for a binomial distribution is given by: For , the probabilities are: Step 2: Apply the given condition.
We are given that . Substituting the probabilities: Simplify: Cancel from both sides (assuming ): Divide by (assuming ): Simplifying: Step 3: Conclusion.
Thus, the value of is .

Concept:
Step 1: Find conjugate.
Step 2: Find modulus.
Step 3: Apply property.
Step 4: Conclusion. Answer = 5

Concept: This problem follows the Binomial Distribution. If probability of success = and probability of failure = , then probability of exactly successes in trials is:

Step 1: Identify the parameters.

Step 2: Apply the binomial probability formula.

Step 3: Simplify the expression. Thus,

Concept: For a binomial distribution, where
• = number of trials
• = probability of success
•

Step 1: Find the value of .

Step 2: Substitute into the variance formula. Thus the variance is