Pick a box, any box

Discrete Mathematics Level 4

A room contains 150 boxes. The boxes are identical except for colour. Each box contains a card which has one of three messages printed on it.

In 78 of the boxes the card says “No Prize”. In 66 of the boxes the card says “Winner $20”. And in 6 of the boxes the card says “Winner $100”. Contestants have been assigned a number corresponding to when they will make their selection.

On a turn, each contestant gets to randomly choose 2 boxes. Once a box is chosen it is removed from the room.

You are the second contestant.

If the probability that you will select at least one of the boxes containing a $100 prize can be expressed as \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).


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