Math Problem Statement
A box contains 2 n tickets of which nCr
tickets bear the number r (r = 0,1,2,3, ... , n). Two tickets
are drawn from the box. Find the expectation of the sum of their numbers.
Solution
We are tasked with finding the expectation of the sum of two ticket numbers when two tickets are drawn from a box containing tickets. The ticket numbers are distributed in a way that tickets bear the number , where . Here's how we can calculate the expectation.
Step 1: Define the random variables
Let the two random variables and represent the numbers on the two tickets drawn. The total number of possible tickets is , and we are interested in the expected value of the sum .
Thus, the goal is to find: Since the tickets are drawn randomly, and are identically distributed, and: Hence, we first calculate , the expectation of a single ticket number.
Step 2: Calculate the expectation of
The probability that a ticket drawn bears the number is proportional to how many tickets bear the number , which is . The total number of tickets is (since ).
The expectation of a single ticket number is given by: The probability of drawing a ticket with number is: Thus, the expectation becomes:
Step 3: Simplify the expectation expression
This summation can be simplified using the known result for the expected value of a binomial distribution: This follows because the sum represents the expectation of a binomial distribution , whose mean is .
Step 4: Calculate the expectation of the sum
Now that we have , the expected value of the sum of the two ticket numbers is:
Final Answer:
The expected value of the sum of the numbers on the two tickets drawn is .
Would you like further details on any step?
Here are some related questions to deepen your understanding:
- How do you compute the expectation for more than two tickets drawn?
- What if the number of tickets with number followed a different distribution?
- How does this problem change if tickets are drawn with replacement?
- How would the calculation differ if was replaced by a different function of ?
- How can variance be computed for the sum of the ticket numbers?
Tip: The binomial distribution is often used in expectation problems because its properties allow for clean and simple calculations of mean and variance.
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Math Problem Analysis
Mathematical Concepts
Probability
Expectation
Binomial Distribution
Formulas
E(X1 + X2) = E(X1) + E(X2)
P(X1 = r) = nCr / 2^n
E(X1) = sum(r * P(X1 = r))
E(X1) for Binomial Distribution B(n, 1/2) = n/2
Theorems
Expectation Property of Identically Distributed Variables
Binomial Distribution
Suitable Grade Level
Undergraduate/Advanced High School
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