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Answer
It's absolutely possible to apply multiple probes for containers within the pod. What happens next depends on a probe.
There are three probes listed in Containers probes which can be used: liveness, readiness and startup. I'll describe more about liveness and readiness:
Liveness
livenessProbe: Indicates whether the container is running. If the
liveness probe fails, the kubelet kills the container, and the
container is subjected to its restart policy. If a Container does not
provide a liveness probe, the default state is Success
The kubelet uses liveness probes to know when to restart a container.
For example, liveness probes could catch a deadlock, where an
application is running, but unable to make progress. Restarting a
container in such a state can help to make the application more
available despite bugs.
In case of livenessProbe fails, kubelet will restart the container in POD, the POD will remain the same (its age as well).
Also it can be checked in container events, this quote is from Kubernetes in Action - Marko Lukša
I’ve seen this on many occasions and users were confused why their
container was being restarted. But if they’d used kubectl describe,
they’d have seen that the container terminated with exit code 137 or
143, telling them that the pod was terminated externally
Readiness
readinessProbe: Indicates whether the container is ready to respond to
requests. If the readiness probe fails, the endpoints controller
removes the Pod's IP address from the endpoints of all Services that
match the Pod. The default state of readiness before the initial delay
is Failure. If a Container does not provide a readiness probe, the
default state is Success
The kubelet uses readiness probes to know when a container is ready to
start accepting traffic. A Pod is considered ready when all of its
containers are ready. One use of this signal is to control which Pods
are used as backends for Services. When a Pod is not ready, it is
removed from Service load balancers.
What happens here is kubernetes checks if webserver in container is serving requests and if not, readinessProbe fails and POD's IP (generally speaking entire POD) will be removed from endpoints and no traffic will be directed to the POD.
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