A magic shop sells a coin which is rather biased. It comes up heads 40% of the time. Suppose the shop owner puts on a demonstration his customers by flipping the coin 12 times. For each item below, write the letter which corresponds to the correct value. Some letters may not be used and some letters may be used more than once.

• Probability of exactly 6 heads?

• Probability of exactly 9 tails?

• Probability of more than than 2 heads?

• Probability of at least 5 heads?

• Probability of less than 3 heads?

• Mean/Expected value for the number of heads?

• Variance for the number of heads?

Choices: A. 0.012 B. 0.142 C. 0.158 D. 0.177 E. 0.775 F. 0.917 G. 1.70 H. 2.88 I. 4.80 J. None of the above.

Steele Electronics Inc. sells expensive brands of stereo equipment in several shopping malls. The marketing research department of Steele reports that 35% of the customers entering the store that indicate they are browsing will, in the end, make a purchase. Let the last 20 customers who enter the store be a sample.

a. How many of these customers would you expect to make a purchase? (Round the final answer to the nearest whole number.)

Number of Customers            

b. What is the probability that exactly six of these customers make a purchase? (Round the final answer to 4 decimal places.)

Probability            

c. What is the probability 11 or more make a purchase? (Round the final answer to 4 decimal places.)

Probability            

d. Does it seem likely at least one will make a purchase ("likely" refers if the probability is more than 70%)?

(Click to select)  Yes  No

An insurance company found that 25% of all insurance policies are terminated before their maturity date. Assume that 10 polices are randomly selected from the company’s policy database. Out of the 10 randomly selected policies;

a) What is the expected number of policies to be terminated before maturing? [1]

b) What is the standard deviation for the number of policies terminated before maturity? [1]

c) What is the probability that no policy will be terminated before maturity? [1]

d) What is the probability that all policies will be terminated before maturing? [2]

e) What is the probability that at least two policies will be terminated? [3]

f) What is the probability that more than 5 but less than eight policies will be terminated? [3] g) What is the probability that at most eight policies will be terminated? [3]

The number of female customers arriving to a coffee shop follow a Poisson process with a mean rate of 3 per hour. The number of male customers arriving to the same coffee shop also follow a Poisson process with a mean rate of 6 per hour and their arrival is independent of the arrivals of female customers.

a) What is the probability that the next customer will arrive within 5 minutes?

b) What is the probability that exactly thee customers will arrive in the next 5 minutes?

c) What is the probability of exactly two male and exactly one female customer will arrive in the next 5 minutes?

d). Parts b) and c) ask for the probability of exactly three arrivals in the next 5 minutes. Are they identical? Explain why?

e) What is the probability of exactly five customers will arrive between 6 and 7 hours from now?

Task 1: Roulette wheel simulation

A roulette wheel has 38 slots of which 18 are red, 18 are black, and 2 are green. If a ball spun on to the wheel stops on the color a player bets, the player wins. Consider a player betting on red. Winning streaks follow a Geometric(p = 20/38) distribution in which we look for the number of red spins in a row until the first black or green. Use the derivation of the Geometric distribution from the Bernoulli distribution to simulate the game. Namely, generate Bernoulli(p = 20/38) random variates (0 = red; 1 = black or green) until a black or green occurs.

Code set-up

A while loop allows us to count the number of spins until a loss. If we use indicator variable lose to note a win (1) or loss (0), the syntax is “while we have not lost (i.e., lose==0), keep spinning.” Once you win, the while loop ends and the variable streak has counted the number of spins. Try running a few times.

streak = 0
lose = 0
p = 20/38
while(lose==0){
lose = (runif(1) < p) # generate Bernoulli with probability p
streak = streak + 1 # tally streak
}
streak

## [1] 2

The problem

The code chunk above performs the experiment once: spin the roulette wheel until you lose and record the number of spins. Simulate 1000 experiments. As usual, do this by wrapping the code chunk above within a for-loop and storing the number of spins streak in a vector.

# [Place code here]

Report the following:

• Histogram of the win streak length. Note that this is a discrete distribution so should place histogram bars at discrete values {0, 1, 2, …}. This may be done with the breaks option within hist. If your storage variable is called winstreak:

hist(winstreak, br=seq(min(winstreak)-0.5, max(winstreak+0.5)), main="")

• Average length of the win streak. [Answer here]
• Standard deviation of the winning streak lengths. [Answer here]
• Compare the empirical average and standard deviation in the previous two bullets to the true values from the Geometric(p = 20/38) distribution.

[Answer here]

• Longest winning streak. [Answer here]

According to a Yale program on climate change communication survey, 71% of Americans think global warming is happening.†

(a)

For a sample of 16 Americans, what is the probability that at least 13 believe global warming is occurring? Use the binomial distribution probability function discussed in Section 5.5 to answer this question. (Round your answer to four decimal places.)

(b)

For a sample of 140 Americans, what is the probability that at least 90 believe global warming is occurring? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.)

(c)

As the number of trials in a binomial distribution application becomes large, what is the advantage of using the normal approximation of the binomial distribution to compute probabilities?

As the number of trials becomes large, the normal approximation simplifies the calculations required to obtain the desired probability.As the number of trials becomes large, the normal approximation gives a more accurate answer than the binomial probability function.   

(d)

When the number of trials for a binomial distribution application becomes large, would developers of statistical software packages prefer to use the binomial distribution probability function shown in Section 5.5 or the normal approximation of the binomial distribution discussed in Section 6.3? Explain.

Is it better to have the reputation of being powerful or of being fair? For example, do you want to be insured by a company that always wins, or by an insurance company that is willing to allow the benefit of any doubt?

‘Odds’ in horserace betting are defined as follows: 3/1 (three-to-one against) means a horse is expected to win once for every three times it loses; 3/2 means two wins out of five races; 4/5 (five to four on) means five wins for every four defeats, etc.

(a) Translate the above odds into ‘probabilities’ of victory.

(b) In a three-horse race, the odds quoted are 2/1, 6/4, and 1/1. What makes the odds different from probabilities? Why are they different?

(c) Discuss how much the bookmaker would expect to win in the long run at such odds, assuming each horse is backed equally.