Question

select the correct answer. in a factory, the chance that a certain machine works without overheating in the morning is 50%. if it runs smoothly all morning, then there is an 85% chance that it will continue for the rest of the day and a 15% chance that it will stop due to overheating. what is the probability that on a given day it will work in the morning and overheat later on? a. 40% b. 37.5% c. 7.5% d. 3.75%

Explanation:

Step1: Define the probabilities

Let $P(\text{works in morning}) = 50\%=0.5$, $P(\text{stops due to overheating}|\text{works in morning}) = 15\% = 0.15$, $P(\text{continues for rest of day}|\text{works in morning})=1 - P(\text{stops due to overheating}|\text{works in morning})=1 - 0.15 = 0.85$.

Step2: Calculate the probability

We want to find the probability that the machine works in the morning and continues for the rest of the day. By the multiplication - rule of conditional probability $P(A\cap B)=P(A)\times P(B|A)$. Here, $A$ is the event that the machine works in the morning and $B$ is the event that it continues for the rest of the day. So $P(\text{works in morning and continues})=P(\text{works in morning})\times P(\text{continues for rest of day}|\text{works in morning})=0.5\times0.85 = 0.425 = 42.5\%$.

Answer:

There is no correct option provided in the given choices as the answer is $42.5\%$.