A batch of 8 bags of cement is sampled. The probability that a bag weighs less than 50 kg is 0.12. What is the probability that exactly half of the bags weigh less than 50 kg?

Here's how to solve the probability problems:

Part (a)
Let $X$ be the number of bags found to be less than $50 \text{ kg}$. This is a binomial distribution problem, where $n=8$ (number of trials) and $p=0.12$ (probability of success, i.e., a bag being less than $50 \text{ kg}$). The probability mass function is $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$.

a) (i) Exactly half

Step 1: Identify the parameters for the binomial distribution.
$n = 8$
$p = 0.12$
$1-p = 0.88$
We need to find the probability that exactly half of the bags are less than $50 \text{ kg}$, so $k = \frac{8}{2} = 4$.

Step 2: Apply the binomial probability formula.

$$P(X=4) = \binom{8}{4} (0.12)^4 (0.88)^{8-4}$$ $$P(X=4) = \frac{8!}{4!(8-4)!} (0.12)^4 (0.88)^4$$ $$$$