To determine the probability of a patient having a temperature above [tex]$100^{\circ} F$[/tex] or having an infection/illness, we need to analyze the given data step by step.
First, let's summarize the provided data:
- Patients with temperature > [tex]$100^{\circ} F$[/tex] and who had an infection/illness: [tex]\(44\)[/tex]
- Patients with temperature > [tex]$100^{\circ} F$[/tex] and who did not have an infection/illness: [tex]\(11\)[/tex]
- Patients with temperature [tex]\(\leq 100^{\circ} F\)[/tex] and who had an infection/illness: [tex]\(15\)[/tex]
- Patients with temperature [tex]\(\leq 100^{\circ} F\)[/tex] and who did not have an infection/illness: [tex]\(43\)[/tex]
Step 1: Calculate the total number of patients.
[tex]\[
\text{Total patients} = 44 + 11 + 15 + 43 = 113
\][/tex]
Step 2: Calculate the probability of having a temperature above [tex]$100^{\circ} F$[/tex] (Event A).
[tex]\[
P(\text{Temp} > 100^{\circ} F) = \frac{\text{Number of patients with temp > 100^{\circ} F}}{\text{Total patients}}
\][/tex]
Where:
[tex]\[
\text{Number of patients with temp > 100^{\circ} F} = 44 + 11 = 55
\][/tex]
[tex]\[
P(\text{Temp} > 100^{\circ} F) = \frac{55}{113} = 0.487
\][/tex]
Step 3: Calculate the probability of having an infection/illness (Event B).
[tex]\[
P(\text{Infection/Illness}) = \frac{\text{Number of patients with infection/illness}}{\text{Total patients}}
\][/tex]
Where:
[tex]\[
\text{Number of patients with infection/illness} = 44 + 15 = 59
\][/tex]
[tex]\[
P(\text{Infection/Illness}) = \frac{59}{113} = 0.522
\][/tex]
Step 4: Calculate the probability of having both a temperature above [tex]$100^{\circ} F$[/tex] and an infection/illness (Intersection of A and B).
[tex]\[
P(\text{Temp} > 100^{\circ} F \cap \text{Infection/Illness}) = \frac{\text{Number of patients with temp > 100^{\circ} F and infection/illness}}{\text{Total patients}}
\][/tex]
Where:
[tex]\[
\text{Number of patients with temp > 100^{\circ} F and infection/illness} = 44
\][/tex]
[tex]\[
P(\text{Temp} > 100^{\circ} F \cap \text{Infection/Illness}) = \frac{44}{113} = 0.389
\][/tex]
Step 5: Calculate the probability of either having a temperature above [tex]$100^{\circ} F$[/tex] or having an infection/illness (Union of A and B).
Using the formula for the union of two events:
[tex]\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\][/tex]
[tex]\[
P(\text{Temp} > 100^{\circ} F \text{ or Infection/Illness}) = 0.487 + 0.522 - 0.389 = 0.619
\][/tex]
Therefore, the probability [tex]\( P(\text{Temp} > 100^{\circ} F \text{ or Infection/Illness}) \)[/tex] is [tex]\( \boxed{0.619} \)[/tex].
