A canal system is shown in the figure. Water flows from A to B through two channels. Gates $G_1$ and $G_2$ are operated independently to regulate the flow. Probability of $G_1$ to be open is 10% while that of $G_2$ is 20%. The probability that water will flow from A to B is
To calculate the probability of water flowing from A to B, we need to consider the scenarios in which the gates $G_1$ and $G_2$ are open because water can only flow if at least one gate is open.
Given:
- Probability of $G_1$ being open, \(P(G_1) = 0.10\)
- Probability of $G_2$ being open, \(P(G_2) = 0.20\)
The probability that both gates are closed is calculated as follows:
- Probability that $G_1$ is closed = \(1 - P(G_1) = 0.90\)
- Probability that $G_2$ is closed = \(1 - P(G_2) = 0.80\)
Therefore, the probability that both gates are closed is the product of both individual probabilities:
\(P(\text{Both Closed}) = 0.90 \times 0.80 = 0.72\)
Thus, the probability that at least one gate is open, allowing water to flow, is:
\(P(\text{At least one open}) = 1 - P(\text{Both Closed}) = 1 - 0.72 = 0.28\)
So, the probability that water will flow from A to B is 28%.
Therefore, the correct answer is 28%.
Important Questions from Probability (Notes)
- A stick of length L is broken into two pieces at random. What is the average length of the smaller piece?
- A fair die was thrown three times and the outcome was repeatedly six. If the die is thrown again what is the probability of getting six?
- A standard fair die is rolled until some face other than 5 or 6 turns up. Let $X$ denote the face value of the last roll, and $A = \{X \text{ is even}\}$ and $B = \{X \text{ is at most 2}\}$. Then,
Consider a Markov chain having state space $S = \{1,2,3,4\}$ with transition probability matrix $P = (p_{i,j})$ given by
$$\begin{matrix} & \begin{matrix} 1 & \quad 2 & \quad 3 & \quad 4 \end{matrix} \\ \begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix} & \begin{bmatrix} 1/2 & 0 & 1/2 & 0 \\ 1/4 & 1/4 & 1/4 & 1/4 \\ 1/3 & 0 & 1/3 & 1/3 \\ 1/2 & 0 & 1/2 & 0 \end{bmatrix} \end{matrix}$$
Then- Let $X$ and $Y$ be two random variables with joint probability density function $f(x, y) = \begin{cases} \frac{1}{\pi} & \text{if } 0 \leq x^2 + y^2 \leq 1 \\ 0 & \text{otherwise.} \end{cases}$ Which of the following statements are correct?
