Question

Math

Posted 4 months ago

`Suppose X is a discrete random variable on the set of positive integers such that for each positive integer n, the probability that X = n is 1/2^n. If Y is a random variable with the same probability distribution and X and Y are independent, what is the probability that the value of at least one of the variables X and Y is greater than 3 ?`

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Calculate the probability that X is less than or equal to 3. This is the sum of the probabilities that X equals 1, 2, or 3

step 2

The probability that X is less than or equal to 3 is $P(X \leq 3) = \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3}$

step 3

Simplify the sum to find $P(X \leq 3) = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{7}{8}$

step 4

Since X and Y are independent, the probability that both X and Y are less than or equal to 3 is $P(X \leq 3 \text{ and } Y \leq 3) = P(X \leq 3) \times P(Y \leq 3)$

step 5

Calculate this probability: $P(X \leq 3 \text{ and } Y \leq 3) = \left(\frac{7}{8}\right)^2$

step 6

Simplify the expression to find $P(X \leq 3 \text{ and } Y \leq 3) = \frac{49}{64}$

step 7

The probability that at least one of X or Y is greater than 3 is the complement of the probability that both are less than or equal to 3

step 8

Calculate this complement: $P(\text{at least one } > 3) = 1 - P(X \leq 3 \text{ and } Y \leq 3)$

step 9

Substitute the previously found probability: $P(\text{at least one } > 3) = 1 - \frac{49}{64}$

step 10

Simplify to find the final probability: $P(\text{at least one } > 3) = \frac{15}{64}$

Answer

The probability that the value of at least one of the variables X and Y is greater than 3 is $\frac{15}{64}$.

Key Concept

Complement of a Probability

Explanation

The probability that at least one of two independent events occurs is the complement of the probability that neither occurs. By calculating the probability that both X and Y are less than or equal to 3 and subtracting from 1, we find the probability that at least one is greater than 3.

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