Well, the probability of a tire being defective is 8%, so the probability of a tire not being defective is 92%.
To find the probability that in a sample of 1,600 tires, 150 or fewer will be defective, we need to add up the probabilities of having 0, 1, 2, 3, ..., 150 defective tires.
However, calculating all those probabilities individually can be a bit tedious, so let me tell you a joke to lift your spirits!
Why did the tire go to the doctor?
Because it was feeling a little flat!
Now, let's get back to the problem. To calculate the probability, we can use a binomial distribution. The formula is:
P(X ≤ k) = Σ (n C k) * p^k * q^(n-k)
Where:
- P(X ≤ k) is the probability of getting k or fewer defective tires in a sample of n tires.
- n is the number of trials (in this case, the number of tires we sample, which is 1,600).
- k is the number of successful events (defective tires) we're interested in (up to 150 in this case).
- (n C k) is the binomial coefficient, which represents the number of ways to choose k out of n items.
- p is the probability of success (a tire being defective), 8% in this case.
- q is the probability of failure (a tire not being defective), which is 92%.
So, using this formula, you can calculate the probability that in a sample of 1,600 tires, 150 or fewer will be defective. Just remember to include the binomial coefficient for each value of k. Good luck with your calculations!