A construction firm purchased 3 tractors from a certain company. At the end of the fifth year, let $E_1$, $E_2$, $E_3$ denote, respectively, the events that tractors no. 1, 2, and 3 are still in good operational condition.
(a) Define the following events at the end of the 5th year, in terms of $E_1$, $E_2$, and $E_3$, and their respective complements:
- A = only tractor no. 1 is in good condition.
- B = exactly one tractor is in good condition.
- C = at least one tractor is in good condition.
$A = E_1 \cap \overline{E_2} \cap \overline{E_3}$
$B = (E_1 \cap \overline{E_2} \cap \overline{E_3}) \cup (\overline{E_1} \cap E_2 \cap \overline{E_3}) \cup (\overline{E_1} \cap \overline{E_2} \cap E_3)$
$C = E_1 \cup E_2 \cup E_3$
Past experience indicates that the chance of a given tractor manufactured by this company having a useful life longer than 5 years (i.e., in good condition at the end of the 5th year) is 60%. If one tractor needs to be replaced (not in good operational condition) at the end of the 5th year, the probability of replacement for one of the other two tractors is 60%; if two tractors need to be replaced, the probability of replacement of the remaining one is 80%.
(b) Evaluate the probabilities of the events A, B, and C.
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My problem starts with (b). It looks like there's a bunch of conditional probabilities, but I'm not sure how to begin parsing the prompt. It seems like the prompt is suggesting that:
$P(\neg E_a | \neg E_b \cap \neg E_c) = 0.80$, where a, b, and c can be uniquely 1, 2, or 3 in any order.
I'm not sure how to interpret the following:
If one tractor needs to be replaced (not in good operational condition) at the end of the 5th year, the probability of replacement for one of the other two tractors is 60%
My best guess is this:
$P(\neg E_a | E_b \cap \neg E_c) = 0.60$
Otherwise, $P(E_{a=b=c}) = 0.6$.
My interpretation of the problem might be wrong, but even so, I still am not sure how to continue from here.