Much easier: just have it trigger when the probe goes well above body temperature, in F that would be 110-120 or so. A hair dryer should have no problem producing that kind of heat.
Then I just looked up your sensors as I don't know the types: they're based around an NTC. You can also simply wire up a voltage divider of a 10k NTC with 10k fixed resistor, connect the mid point to an analog port, and do a quick calibration on at which value to trigger your relay. Easy, cheap and very reliable.
Code for this is dead easy:
void loop() {
if (analogRead(A0) > thresholdValue) { // can be greater or less than a value - depends on wiring and NTC and required temperatures)
digitalWrite(relayPin, HIGH); // relay on.
delay(5000); // Wait for 5 seconds.
digitalWrite(relayPin, LOW); // relay off.
delay(600000); // 10 minutes wait to let it cool down again.
}
}