While you are away on vacation, you ask a neighbor to water your ailing plant. If the plant does not receive water, it will die with a probability of 0.85. With water it will die with probability 0.5. You are 90 % certain the neighbor will remember to water the plant. You come back from the vacation and the plant is dead. What is the probability that the plant died because neighbor forgot to water it?

Question

Sally Cresswell · Accepted Answer

Let  A={plant died},and  H1={neighbour watered the plant},  H2={neighbour did not water the plant}  We need to find P(H2∣A).  We will use the Bayes&#039; Theorem (extended form):    P(H2∣A)=P(H2)PA∣H2P(H1)P(A∣H1)+P(H2)P(A∣H2)   =0.1⋅0.85(0.9⋅0.5)+(0.1⋅0.85)=0.0850.535  Therefore,  P(H2∣A)=17107

nick1337 · Accepted Answer

Result:0.159 or 15.9%Solution:To solve the given problem, we can apply Bayes&#039; theorem. Let&#039;s define the following events:A: The plant dies because the neighbor forgot to water it.B: The plant dies.We need to find the conditional probability P(A|B), which represents the probability that the plant died because the neighbor forgot to water it, given that the plant is dead.According to Bayes&#039; theorem, we have:P(A|B)=P(B|A)·P(A)P(B)To calculate this, we need to determine the individual probabilities:P(B|A): The probability that the plant dies given that the neighbor forgot to water it. This is given as 0.85.P(A): The probability that the neighbor forgot to water the plant. This is given as 1 - 0.9 (since the probability that the neighbor remembers to water the plant is 90%).P(B): The overall probability that the plant dies.To calculate P(B), we can use the law of total probability:P(B)=P(B|A)·P(A)+P(B|¬A)·P(¬A)Here, ¬A represents the event that the neighbor remembered to water the plant.P(B|¬A): The probability that the plant dies given that the neighbor remembered to water it. This is given as 0.5.P(¬A): The probability that the neighbor remembered to water the plant, which is the complement of P(A) (i.e., P(¬A)=1−P(A)).Now, let&#039;s substitute the given probabilities into the formulas:P(B)=(0.85)·(1−0.9)+(0.5)·(0.9)Simplifying further:P(B)=0.85·0.1+0.5·0.9P(B)=0.085+0.45P(B)=0.535Now, we can calculate P(A|B) using Bayes&#039; theorem:P(A|B)=0.85·(1−0.9)0.535P(A|B)=0.85·0.10.535P(A|B)=0.0850.535P(A|B)≈0.159Therefore, the probability that the plant died because the neighbor forgot to water it is approximately 0.159 or 15.9%.

RizerMix · Accepted Answer

Step 1: We need to find the probability P(A|B), which represents the probability that the plant died because the neighbor forgot to water it, given that the plant is dead.According to Bayes&#039; theorem, we have:P(A|B)=P(B|A)·P(A)P(B)Step 2: Now let&#039;s calculate each of these probabilities step by step:P(B|A) represents the probability that the plant dies (event B) given that the neighbor forgot to water it (event A). In this case, the plant will die with a probability of 0.85. Therefore, P(B|A)=0.85.P(A) represents the probability that the neighbor forgot to water the plant (event A). We are given that we are 90% certain that the neighbor will remember to water the plant, so the probability that the neighbor forgets is 1−0.9=0.1. Hence, P(A)=0.1.P(B) represents the probability that the plant dies (event B). This can happen either because the neighbor forgot to water it (event A) or because the neighbor watered it but the plant still died. We can calculate this probability using the law of total probability:P(B)=P(B|A)·P(A)+P(B|¬A)·P(¬A)Here, ¬A represents the event that the neighbor remembered to water the plant. If the neighbor remembered to water the plant, the plant will die with a probability of 0.5. We are 90% certain that the neighbor will remember, so P(B|¬A)=0.5 and P(¬A)=0.9. Substituting these values, we get:P(B)=0.85·0.1+0.5·0.9P(B)=0.085+0.45P(B)=0.535Step 3: Now we can substitute these values back into Bayes&#039; theorem:P(A|B)=0.85·0.10.535P(A|B)=0.0850.535P(A|B)=17107Therefore, the probability that the plant died because the neighbor forgot to water it is 17/107.

Vasquez · Accepted Answer

A: The plant dies because the neighbor forgot to water it.B: The plant dies.C: The neighbor remembered to water the plant.We are given the following probabilities:P(A|BC)=0.85 (The plant dies without water)P(A|BC)=0.5 (The plant dies with water)P(C)=0.9 (The neighbor remembers to water the plant)We need to find P(A|B) (The probability that the plant died because the neighbor forgot to water it).Using Bayes&#039; theorem, we can write:P(A|B)=P(B|A)·P(A)P(B)We can calculate the individual probabilities as follows:P(B|A)=1 (If the neighbor forgot to water, the plant will die)P(A)=P(A|BC)·P(BC)+P(A|BC)·P(BC)P(BC) can be calculated as the complement of P(B):P(BC)=1−P(B)=1−P(B|AC)·P(AC)−P(B|AC)·P(AC)Now, substituting the values and calculating:P(B|A)=1P(A)=0.85·(1−P(B|AC)·P(AC)−P(B|AC)·P(AC))+0.5·P(BC)P(B)=P(B|A)·P(A)+P(B|AC)·P(AC)=1·P(A)+0·P(AC)=P(A)Therefore, we have:P(A|B)=P(B|A)·P(A)P(B)=1·P(A)P(A)=1Thus, the probability that the plant died because the neighbor forgot to water it is 100%.

You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.85. With water it will die w

Answered question

Answer & Explanation

New Questions in High school probability