Answer:
0.9894 = 98.94% probability that he does not have a TBI.
Step-by-step explanation:
Conditional Probability
We use the conditional probability formula to solve this question. It is
In which
P(B|A) is the probability of event B happening, given that A happened.
is the probability of both A and B happening.
P(A) is the probability of A happening.
In this question:
Event A: Negative screen
Event B: Does not have a TBI.
Probability of a negative screen:
93 are negative and do not have a TBI.
1 is negative and has a TBI.
Out of 112.
So
![P(A) = \frac{93+1}{112} = \frac{94}{112}](https://tex.z-dn.net/?f=P%28A%29%20%3D%20%5Cfrac%7B93%2B1%7D%7B112%7D%20%3D%20%5Cfrac%7B94%7D%7B112%7D)
Probability of a negative screen and not having a TBI:
93 are negative and do not have a TBI, out of 112, so:
![P(A \cap B) = \frac{93}{112}](https://tex.z-dn.net/?f=P%28A%20%5Ccap%20B%29%20%3D%20%5Cfrac%7B93%7D%7B112%7D)
One of the veterans has a negative screen and wants to know the probability that he does not have a TBI.
![P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{93}{112}}{\frac{94}{112}} = \frac{93}{94} = 0.9894](https://tex.z-dn.net/?f=P%28B%7CA%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28A%29%7D%20%3D%20%5Cfrac%7B%5Cfrac%7B93%7D%7B112%7D%7D%7B%5Cfrac%7B94%7D%7B112%7D%7D%20%3D%20%5Cfrac%7B93%7D%7B94%7D%20%3D%200.9894)
0.9894 = 98.94% probability that he does not have a TBI.