How do market makers avoid losses in their stock inventory?

Some market makers do probably have positive overall exposure to the market, but they can do a lot to avoid this situation.

1. They make markets in many different assets at the same time, so they don't have too much concentrated risk in any one asset. That is, their inventory becomes a large, diversified basket similar to the overall market.
2. They actively monitor their overall market exposure. If they are long the market, they can take on offsetting positions (such as taking a short position in a market index future).

Market makers employ risk managers that are constantly looking at the overall market risk, the sector risk, the counterparty risk, the liquidity risk, and many other types of risks the firm is exposed to. They compare these risk levels to the tolerances defined by the chief risk officer of the firm and board of directors. Thousands of highly skilled and educated people across many firms are engaged in this activity (risk management). They have lots of hedging tools at their disposal to reduce exposure when they near or exceed these levels. However, since hedging is often costly, so some firms may maintain some risk most of the time.