Final answer:
To find the probabilities, we can use the binomial probability formula: P(x) = (nCx) * p^x * (1-p)^(n-x), where n is the number of trials, x is the number of successful outcomes, p is the probability of success, and (nCx) is the number of combinations. For part (a), the probability exactly 7 tires are defective is 10.64%. For part (b), the probability of at least 7 tires being defective is 10.71%.
Step-by-step explanation:
To find the probabilities, we can use the binomial probability formula:
P(x) = (nCx) * p^x * (1-p)^(n-x)
where n is the number of trials, x is the number of successful outcomes, p is the probability of success, and (nCx) is the number of combinations.
a) To find the probability that exactly 7 tires are defective, we have n = 8, x = 7, p = 0.3:
P(7) = (8C7) * (0.3^7) * (0.7^1) = 8 * 0.3^7 * 0.7 = 0.1064 or 10.64%
b) To find the probability that at least 7 tires are defective, we need to find the probabilities of 7, 8 tires being defective and add them:
P(at least 7) = P(7) + P(8) = 0.1064 + (8C8) * (0.3^8) * (0.7^0) = 0.1064 + 0.3^8 = 0.1064 + 0.000657 = 0.107057 or 10.71%