1. A marketing agency has developed its vacation packages to promote a timeshare plan at a new resort. It estimates that 20% of potential customers will choose the Day Plan, which does not include overnight accommodations; 40% will choose the Overnight Plan, which includes one night at the resort; and 40% will choose the Weekend Plan, which includes two nights. Let X be a random variable for the number of nights that a potential customer will choose to stay.
2. (a)Give the probability distribution for X.
3. (b)Compute the expected value for the number of nights that customers will stay.
4. (c)Compute the standard deviation for the number of nights that customers will stay.
5. (d)What is the probability that in a sample of 10 independently selected customers, exactly three will choose the Day Plan? Make sure to define an appropriate random variable.
6. Of the 2500 accounts of a particular bank, 125 have been fraudulently altered. The alterations are sufficiently subtle that only a detailed audit can uncover them. Fifty business accounts are chosen at random for detailed auditing. Let X be a random variable for the number of fraudulently altered accounts in the sample of fifty.
7. (a) ExplainwhyXfollowsahypergeometricdistribution,andgivetheparametersofthedistribution. (b) Find the probability that the sample will contain at least two accounts that have been altered. (c) Can the distribution for X be approximated by the binomial distribution? Explain.

(d) Regardless of your answer in (c), use the binomial distribution to approximate the probability in (b).

3. Logging trucks that are driven fast over very rough, temporary roads will experience tire failures due to cuts, blowouts, and large punctures. Such failures occur according to a Poisson distribution with a mean rate of 4 per 10,000 km. For each part of this question, make sure to define an appropriate random variable.

(a) If a truck is driven for 10,000 km, what is the probability of no more than 2 tire failures?

(b) If a truck is driven for only 1,000 km, what is the probability that it does not have any tire failures?