Since the auto body shop receives 70% of its parts from one manufacturer, the probability that a part is not from this manufacturer is 30%.
To find the probability that the first part not from this manufacturer is the 6th part selected, we can use the formula for the probability of a specific order of events in a sequence, which is:
P(event 1 and event 2 and ... and event n) = P(event 1) × P(event 2 | event 1) × P(event 3 | event 1 and event 2) × ... × P(event n | event 1 and event 2 and ... and event n-1)
In this case, let event 1 be selecting a part not from the manufacturer, and let event 2 through event 6 be selecting any part in any order. Then, we have:
P(select 6th part not from manufacturer first) = P(not from manufacturer) × P(any part) × P(any part) × P(any part) × P(any part) × P(any part | event 1 and event 2 and event 3 and event 4 and event 5)
Since the events of selecting each part are independent and the probability of selecting any part is the same, we can simplify this expression to:
P(select 6th part not from manufacturer first) = 0.3 × 0.7^5
Calculating this expression gives:
P(select 6th part not from manufacturer first) = 0.3 × 0.16807 = 0.05042
Therefore, the probability that the first part not from this manufacturer is the 6th part selected is approximately 0.05042 or 5.04%.