FLC OF B737-300 HAVE NAV PROBS INFLT WHEN BOTH IRS AND FMC DO NOT UPDATE INFO. THEY REVERT TO RAW DATA AND LATER THE PROB CLRS ITSELF. A GPS IS FOUND TO BE IN USE IN THE CABIN.

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ACFT EQUIPPED WITH IRS AND FMC. I ARRIVED AT ACFT AND COMPLETED PREFLT INCLUDING A FULL ALIGNMENT OF IRS. ONCE AIRBORNE; CONFIRMED IRS/FMC NAV FUNCTIONS WERE WORKING NORMALLY BY USING FIX PAGE AND RAW DATA. APPROX 10-15 MINS INTO THE CLB; BOTH CAPT AND MYSELF NOTED THE FMC WAS NOT AUTO UPDATING ITSELF. ABOUT THIS TIME WE RECEIVED A 'VERIFY POS' MESSAGE FROM THE FMC. WE REVERTED TO RAW DATA FOR NAV AND I CHKED THE IRS AND FMC POS PAGE. BOTH THE L AND R IRS CLAIMED THE ACTUAL POS WAS 3.8 AND 3.4 NM (RESPECTIVELY); 90 DEGS TO OUR L. WE WERE GOING TO COMPLETE A POS SHIFT EXERCISE WHEN WE NOTED WE HAD AUTO UPDATING FUNCTIONS BACK AND THE FMC POS WAS CORRECTING ITSELF. LATER IN THE FLT; A FLT ATTENDANT CALLED AND ASKED OUR GND SPD. I TOLD HER IT WAS 389 KTS. SHE STATED SHE HAD LOST A BET BECAUSE A PAX GUESSED 388 KTS. I ASKED HER TO ASK THE PAX IF HE HAD A GPS. THE PAX SAID YES. I ASKED HER TO TELL THE PAX TO TURN IT OFF AND KEEP IT TURNED OFF. THE FLT CONTINUED UNEVENTFULLY. AT DEPLANING; I FOUND OUT THE PAX WAS EMPLOYED BY ANOTHER AIRLINE. FLT IN QUESTION WAS TO PHX IN A B737-300.