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?
For part a would the solution simply be 0.8^5
and for part b) how would I go about determining 75 % of the movies would it be 187.5 or a whole number 188
For part a, the probability that all 5 copies are returned by Monday can be calculated using the given information that 80% of people return their movies within 2 days. Since the rental policy now allows an unlimited return date, it means that all movies can still be returned within 2 days. Therefore, the probability that a single movie is returned within 2 days is 80%, or 0.8.
To calculate the probability that all 5 copies are returned by Monday, we can multiply the individual probabilities together since the events are independent. So the solution would indeed be 0.8^5, which equals 0.32768, or around 32.77%.
For part b, we need to find the probability that at least 75% of the 250 movies rented on Saturday are returned by Monday. To determine 75% of 250, we multiply 250 by 0.75, which equals 187.5. However, since we cannot have a fraction of a movie, we round this down to the nearest whole number, which gives us 187.
Now, we can find the probability using a binomial distribution. The probability that exactly k movies are returned out of n movies rented can be calculated using the formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
In this case, n = 250 (total movies rented on Saturday), and p = 0.8 (probability of a movie being returned). We want to find the probability that at least 187 movies are returned, so we need to calculate the individual probabilities for k = 187, 188, 189, ..., 250 and sum them up.
Alternatively, you can use a cumulative distribution function (CDF) for the binomial distribution to directly find the probability of having at least 187 successes out of 250 trials. Many statistical software or online calculators can perform this calculation easily.
Please let me know if you would like further assistance with the calculations.