Find The Lady (a tough non-Bridge problem)

You are having a fund-raising day at the Bridge Club and someone suggests a 'find-the-lady' stand.

The stall-holder (dealer) has 5 cards on a table, four deuces and a queen which they place randomly in a line face downwards.

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For \$1 you can choose a card and if you pick the Queen you win a prize.

If you fail you can pay another \$1 to have a second go. Your card is replaced face down and before you choose you must turn your back whilst the dealer swaps the Queen with an adjacent card (the dealer will choose randomly if there is a choice).

You can have as many turns as you like on this basis.

Once someone wins or quits the dealer re-deals the cards randomly.

The prize is a specially printed club pen which cost the club \$3.50 each.

If everyone is as clever as you and uses perfect strategy will the club make any money?

Assume the game is played 160 times (no reason, apart from someone wanted a number!)

(I can't claim any credit for the strategy part of the puzzle, its a re-write of a 'cat-in-a-box' puzzle on Brilliant.org).

The club will make a big profit
The club will make a small profit
The club will make a small loss
The club will make a big loss

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