Well, if the ping results are not really important for you, then you can have only one check_ping as check-host-alive under host definition. I've said not important, because, check-host-alive is not executed on a regular basis, as mentioned in previous post. It is done only when Nagios thinks it is necessary to check the host.
Although check-host-alive can be any of the check-plugins you want, but as default it is done by check-ping.
With both, the check-host-alive as a host check, and a check_ping service, you get ping check results on a regular basis, as defined in check_interval variable in service definition.
Check's won't be duplicated, because Nagios won't execute check-host-alive until services associated with the host start to get Critical or Unknown.
So it is really up to you, if you want check_ping on a regular basis or not.