Does living in the same house with that of a chicken pox patient necessarily lead to contacting that disease?

Yes if never before: All household members are at risk for contracting chicken pox if one member has active chicken pox and these household members have never had chicken pox. If they have contracted chicken pox in the past, then there is theoretically no risk.

Depends: Most adults (>90%) have blood test evidence of prior chickenpox even if they don't remember having the disease. Prior disease gives permanent immunity. If not immune the risk is there. As little as 1/2 hour spent in the same room with a chickenpox patient is enough to transfer the disease.