A certain type of artificial heart has four independent, critical components.Failure of any one of these components is a serious problem that will cause the artificial heart to fail. The 5 year failure rate for each component follows: component 1=.02, component 2=.05, component 3=.04, component 4=.03.
What is the probability that an artificial heart functions for 5 years?
I first solved for what is probability each component wont fail (1-probability will fail). Then multiply all probabilities that wont fail
Either-or probabilities are found by adding the individual probabilities.
Do that for failure rates, then subtract from 1.
To find the probability that an artificial heart functions for 5 years, you first need to calculate the probability that each component doesn't fail. Given that the failure rates (probability of failure) for the four components are as follows:
Component 1: 0.02
Component 2: 0.05
Component 3: 0.04
Component 4: 0.03
The probability that component 1 doesn't fail is 1 minus its failure rate, so it would be 1 - 0.02 = 0.98.
Similarly, the probabilities that component 2, 3, and 4 don't fail would be:
Component 2: 1 - 0.05 = 0.95
Component 3: 1 - 0.04 = 0.96
Component 4: 1 - 0.03 = 0.97
To find the probability that all four components don't fail, you multiply these probabilities together:
0.98 * 0.95 * 0.96 * 0.97 = 0.861528
Therefore, the probability that an artificial heart functions for 5 years is approximately 0.8615, or 86.15% (rounded to two decimal places).