The BA is a vital component of a project. Why, b/c they define, along with the customer, the product to be produced. The BA is the liaison between the customer and the technical team (my experience is IT related). The BA acts as the Product Owner to have a clear understanding of the current and target state so as to act on behalf of the customer to see the development of the product through. I cannot tell you the statistics of the success rates with or without, but I would certainly bet in favor of when the BA is involved.
The BA elicits the requirements. This is not always after the Project Charter is signed, b/c quite honestly, there isn't always a charter. The BA helps to work with the Business group to figure out what the intent is and get enough high-level information so to formulate a narrative to determine the depth of the work. It may be very small, and the requirements can be hashed out, and the work can be done. On the flip side, there are times when the narrative is shared with other groups, such as Enterprise Architecture, to leverage existing capabilities within the organization, or determine what solution is best. There are many reasons for this partnership and is probably another discussion.
During the project lifecycle, the BA is instrumental in working in conjunction with the PM to monitor, consult, and ensure there is a clear understanding of the product from a technical perspective. They are the voice and reason of the Business.
My opinion may be a bit skewed b/c I work so much as both roles.