Solve the following exercises:
1. Using the binomial random distribution probability formula given below, calculate the probabilities based on the provided parameters:
Where:
$P(y) = \binom{n}{y}p^yq^{n-y}$
$\binom{n}{y} = \frac{n!}{y!(n-y)!}$
a. $y = 3, n = 8, and p = 0.35$
b. $y = 5, n = 8, and p = 0.6$
c. $P(3 \le y \le 5)$ when $n = 7$, and $p = 0.6$
d. $P(1 \le y)$ when $n = 9$, and $p = 0.1$
2. The National Transport Safety Center (NTSC) in the Kingdom of Saudi Arabia reported that 7 in 10 auto accidents involve a single vehicle. Suppose 15 accidents are randomly selected. Using the Binomial Cumulative Probability Distribution Table, answer the following questions:
a. What is the probability that at most 4 involve a single vehicle?
b. What is the probability that exactly 4 involve a single vehicle?
c. What is the probability that at least 2 involve a single vehicle?
d. What is the probability that between 2 and 4 involve a single vehicle?
3. In order to process customers' requests, Golden Wings for Travel Services is using land line telephone to receive voice calls and a fax machine to receive fax messages. Suppose that 25% of the incoming calls involve fax messages and consider a sample of 25 incoming calls. Answer the following questions:
a. What is expected number of calls among the 25 that involve a fax message?
b. What is the standard deviation of the number among the 25 calls that involve a fax message?
