To find the probability of selecting either a US spec or a short bolt, we need to calculate the union of the two events: US spec (denoted as US) and short (denoted as S). The probability of the union is denoted as P(US U S).
The formula for the union of two events is:
P(US U S) = P(US) + P(S) - P(US ∩ S)
Given that there are 8 US spec bolts and 6 short bolts, we can calculate the individual probabilities:
P(US) = 8 / (8 + 6) = 8 / 14 = 4 / 7
P(S) = 6 / (8 + 6) = 6 / 14 = 3 / 7
Now, we need to determine the probability of the intersection of the two events, P(US ∩ S). Since a bolt cannot be both US spec and short, the intersection is empty, and therefore P(US ∩ S) = 0.
Plugging in the values, we can calculate the probability of selecting either a US spec or a short bolt:
P(US U S) = P(US) + P(S) - P(US ∩ S)
= 4/7 + 3/7 - 0
= 7/7
= 1
Therefore, the probability of selecting either a US spec or a short bolt is 1, which means it is guaranteed to happen since it includes all the bolts available.