A boutique hotel has only 40 rooms. Records indicate that the occupancy rate is 94% a. What is the probability that on any given night all 40 rooms are occupied?
b. What is the probability that less than 35 rooms are occupied?
c. What is the probability that between 33 and 38 rooms are occupied (inclusive)?
d. Because of regular cancellations, many hotels in other countries adopt an overbooking strategy (note this
is not legal in Australia). This is a risky process as it may result in customers who have booked not having
a room which requires the hotel to re-book them at other hotels nearby and can lead to both financial and
reputational loss.
This hotels manager is convinced the reason occupancy is not 100% is because of cancellations. As such,
she is considering adopting an overbooking strategy. If she was to adopt a strategy of booking 42 rooms
per night, what is the probability that she would end up having customers who had booked, but for whom
she has no room? Show all working.
用excel的方式来列出答案

Question

A boutique hotel has only 40 rooms. Records indicate that the occupancy rate is 94%        a. What is the probability that on any given night all 40 rooms are occupied?
b. What is the probability that less than 35 rooms are occupied?
c. What is the probability that between 33 and 38 rooms are occupied (inclusive)?
d. Because of regular cancellations, many hotels in other countries adopt an overbooking strategy (note this
is not legal in Australia). This is a risky process as it may result in customers who have booked not having
a room which requires the hotel to re-book them at other hotels nearby and can lead to both financial and
reputational loss.
This hotels manager is convinced the reason occupancy is not 100% is because of cancellations. As such,
she is considering adopting an overbooking strategy. If she was to adopt a strategy of booking 42 rooms
per night, what is the probability that she would end up having customers who had booked, but for whom
she has no room? Show all working.
用excel的方式来列出答案

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To calculate the probability that all 40 rooms are occupied, we assume that the occupancy of each room is independent and use the binomial probability formula: $$ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $$ where $ n $ is the number of trials (rooms), $ k $ is the number of successes (occupied rooms), and $ p $ is the probability of success (occupancy rate). For all 40 rooms to be occupied, $ k = 40 $ and $ p = 0.94 $. ‖ ‖ step 2 ⋮ Since we want the probability of all rooms being occupied, we only need to calculate $ p^{40} $ because $ \binom{40}{40} = 1 $ and $ (1-p)^{0} = 1 $. Thus, the probability is: $$ P(X = 40) = 0.94^{40} $$ ‖ ‖ step 3 ⋮ Calculate the probability using the given occupancy rate: $$ P(X = 40) = 0.94^{40} $$ ‖ ∻Answer∻ ⚹ The probability that all 40 rooms are occupied is $ 0.94^{40} $. ⚹ ∻Key Concept∻ ⚹Binomial Probability Formula⚹ ∻Explanation∻ ⚹The binomial probability formula is used to calculate the probability of a specific number of successes in a series of independent trials.⚹ --- ∻Solution by Steps∻ ‖ step 1 ⋮ To find the probability that less than 35 rooms are occupied, we use the cumulative distribution function (CDF) of the binomial distribution: $$ P(X < k) = \sum_{i=0}^{k-1} \binom{n}{i} p^i (1-p)^{n-i} $$ where $ n = 40 $, $ p = 0.94 $, and $ k = 35 $. ‖ ‖ step 2 ⋮ Calculate the sum of probabilities for $ X = 0 $ to $ X = 34 $ using the binomial probability formula. This can be done using Excel's BINOM.DIST function: $$ P(X < 35) = \text{BINOM.DIST}(34, 40, 0.94, TRUE) $$ ‖ ∻Answer∻ ⚹ The probability that less than 35 rooms are occupied is given by Excel's BINOM.DIST function with parameters 34, 40, 0.94, TRUE. ⚹ ∻Key Concept∻ ⚹Cumulative Distribution Function of Binomial Distribution⚹ ∻Explanation∻ ⚹The CDF is used to calculate the probability of obtaining a result less than a certain value in a binomial distribution.⚹ --- ∻Solution by Steps∻ ‖ step 1 ⋮ To calculate the probability that between 33 and 38 rooms are occupied (inclusive), we use the CDF of the binomial distribution: $$ P(a \leq X \leq b) = P(X \leq b) - P(X < a) $$ where $ a = 33 $ and $ b = 38 $. ‖ ‖ step 2 ⋮ Calculate the CDF for $ X \leq 38 $ and for $ X < 33 $ using Excel's BINOM.DIST function: $$ P(33 \leq X \leq 38) = \text{BINOM.DIST}(38, 40, 0.94, TRUE) - \text{BINOM.DIST}(32, 40, 0.94, TRUE) $$ ‖ ∻Answer∻ ⚹ The probability that between 33 and 38 rooms are occupied is given by the difference of Excel's BINOM.DIST function with parameters 38, 40, 0.94, TRUE and 32, 40, 0.94, TRUE. ⚹ ∻Key Concept∻ ⚹Difference of Cumulative Probabilities⚹ ∻Explanation∻ ⚹To find the probability of an inclusive range, we subtract the CDF value at the lower bound from the CDF value at the upper bound.⚹ --- ∻Solution by Steps∻ ‖ step 1 ⋮ To find the probability of overbooking, we calculate the probability that more than 40 rooms are booked when 42 rooms are sold. This is the complement of the probability that 40 or fewer rooms are occupied: $$ P(X > 40) = 1 - P(X \leq 40) $$ ‖ ‖ step 2 ⋮ Calculate the CDF for $ X \leq 40 $ using Excel's BINOM.DIST function: $$ P(X > 40) = 1 - \text{BINOM.DIST}(40, 42, 0.94, TRUE) $$ ‖ ∻Answer∻ ⚹ The probability of having customers who had booked but for whom there is no room when 42 rooms are booked is $ 1 - \text{BINOM.DIST}(40, 42, 0.94, TRUE) $. ⚹ ∻Key Concept∻ ⚹Complement of Cumulative Probability⚹ ∻Explanation∻ ⚹The complement of the cumulative probability is used to calculate the probability of an event occurring beyond a certain point in a distribution.⚹◊What is the probability of overbooking occurring if the hotel adopts a strategy of booking 42 rooms per night?⍭ Generate me a similar question◊

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