ABC Engineering Corp is interested in creating a prototype for remotely delivering supplies to offshore oil rigs in the North Sea using a medium$-$sized robotic craft. A subsystem in the craft has a shaft that rotates at $1860$ rpm and the design problem now is selecting a rolling$-$contact bearing for the shaft.

$(1)$ Raul, an engineering intern in the team, plans on using a $02-50$ deep$-$groove ball bearing to support $2$ kN of radial load which he thinks will provide a life of $20,000$ hours at $99\%$ reliability.

$(2)$ His supervisor, Jodie thinks that using the more expensive $03-50$ cylindrical roller bearing to last $20,000$ hours of service at $99\%$ reliability will increase its radial load carrying capacity from $2$ kN to $8$ kN which will let them increase the payload on the cract.

All such bearings are rated for $L_{10}={10}^6$ revs. You are brought on as a product design consultant. Provide an analysis proving or disproving either or both statements.