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)
- In a box there are 4 white balls and 6 black balls. A ball is drawn at random. If it is white, it is put back along with two more white balls in the box. If it is black, it is put back in the box and then two black balls are thrown out of the box. Now a ball is drawn again at random from the box. Then, what is the probability that it is black?
- A fair coin is tossed three times. Let A be the event of getting exactly two heads and B be the event of getting at most
two tails, then P(A$\cup$B) is: Bag A contains 3 Red and 4 Black balls while Bag B contains 5 Red and 6 Black balls. One ball is drawn at random from one of the bags and is found to be red. Then, the probability that it was drawn from Bag B is
- If we twice flip a balanced coin, what is the probability of getting at least one head?
1. 1/4
2. 2/4
3. 1/6
4. 3/4 Suppose that the random variable X takes on the values: -1, 0, and 2 with probability $\frac{1}{8}$, $\frac{1}{2}$ and $\frac{3}{8}$. Find the expected value of X.
