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Question
Math
Posted 11 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 11 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≤3)=12+122+123 P(X \leq 3) = \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3}
step 3
Simplify the sum to find P(X≤3)=12+14+18=78 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≤3 and Y≤3)=P(X≤3)×P(Y≤3) 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≤3 and Y≤3)=(78)2 P(X \leq 3 \text{ and } Y \leq 3) = \left(\frac{7}{8}\right)^2
step 6
Simplify the expression to find P(X≤3 and Y≤3)=4964 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(at least one >3)=1−P(X≤3 and Y≤3) P(\text{at least one } > 3) = 1 - P(X \leq 3 \text{ and } Y \leq 3)
step 9
Substitute the previously found probability: P(at least one >3)=1−4964 P(\text{at least one } > 3) = 1 - \frac{49}{64}
step 10
Simplify to find the final probability: P(at least one >3)=1564 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 1564\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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