1. A large video rental chain recently decided to change its rental policy, allowing an unlimited
return date instead of requiring that movies be returned the next day. The marketing
team reasoned that this would greatly increase the popularity of their stores, and also that
80% of people would return their movies within 2 days regardless.
a. Suppose that that a specific store has 5 copies of “The Princess Bride”, and rents
them all on a Saturday. What is the probability that all 5 copies are returned by
Monday?
b. The same store rented 250 movies in total that Saturday. What is the probability that
at least 75% of these movies were returned by Monday?
2. A popular casino in Windsor estimates that the chance that someone sitting at one of their
Black Jack tables has a 0.35 probability of winning a given hand. Assume that all games
played are independent.
a. Isaac decides to play until his first win, and bets $50 per game. When he wins, he
will win $100. What is Isaac’s expected net gain/loss from this strategy?
b. Suppose the casino has 100 people playing the same strategy as Isaac. What is
the probability that the casino sees no net gain from these 100 people (that is, that
it sees a net gain of zero or smaller)? Hint: recall that the variance of a geometric
random variable is (1 p)=p2:
3.
1.5 million Canadians participated in a real-time interactive IQ test. Viewers took the test
in the comfort of their own homes, on the internet or with pen and paper, while seven
teams – tattoo artists, millionaires, fitness instructors, surgeons, mayors, talk jocks and
celebrities – exercised their grey matter in CBC’s Toronto studio. Suppose that 25% of
Canadians scored above average (that is, scoring 110 or more). You took a random sample
of n=81 individuals and recorded their IQ scores. Let bp denote the sample proportion
of individuals who scored above average.
a. What is the probability that bp lies between 19% and 22%?
b. Ninety percent of the time, the sample proportion bp is smaller than what number?
c. Would a value of bp = 0:35 be considered unusual? Justify your answer.
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To answer these questions, we can use probability theory and statistical methods. Let's break down each question and explain the steps to find the answers.
1. Rental Policy:
a. To find the probability that all 5 copies are returned by Monday, we need to use the binomial probability formula.
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
In this case, n = 5 (the number of copies), k = 5 (all 5 copies returned), and p = 0.8 (probability of returning within 2 days).
P(X = 5) = (5 choose 5) * 0.8^5 * (1-0.8)^(5-5)
Calculate the values and you will get the probability that all 5 copies are returned by Monday.
b. To find the probability that at least 75% of the movies were returned, we need to use the binomial cumulative distribution function or simulation.
P(X >= 0.75*n) = 1 - P(X < 0.75*n)
Calculate the values and you will get the probability that at least 75% of the movies were returned by Monday.
2. Casino Game:
a. To find Isaac's expected net gain/loss, we need to calculate the expected value (mean) of the game.
Expected Value = (Probability of winning * Amount won) - (Probability of losing * Amount lost)
In this case, the probability of winning is 0.35, the amount won is $100, and the amount lost is $50.
Calculate the expected value and you will get Isaac's expected net gain/loss from this strategy.
b. To find the probability that the casino sees no net gain, we need to calculate the probability of a net gain of zero or smaller for each individual player, and then use the binomial distribution or simulation to find the probability for 100 players.
Probability of no net gain = P(X <= 0)
Calculate the values and you will get the probability that the casino sees no net gain from these 100 people.
3. IQ Test:
a. To find the probability that the sample proportion (bp) lies between 19% and 22%, we need to use the normal distribution approximation.
Calculate the z-scores for 19% and 22%, and find the probability between these z-scores using the standard normal distribution table or a calculator.
b. To find the value of bp that is smaller than the sample proportion 90% of the time, you need to use the normal distribution.
Find the z-score that corresponds to the 90th percentile (0.9) and use it to find the value of bp using the formula:
bp = mean + (z-score * standard deviation)
c. To determine if a value of bp = 0.35 is considered unusual, you need to compare it to the distribution and calculate the z-score.
Calculate the z-score using the formula:
z = (bp - mean) / standard deviation
If the z-score falls outside a certain range (usually beyond -2 to +2), the value is considered unusual.
These explanations should help you understand how to approach and solve the specified problems.