A Myrtle Beach resort hotel has 120 rooms. In the spring...

A Myrtle Beach resort hotel has 120 rooms. In the spring months, hotel room occupancy is approximately $75 \%$ a. What is the probability that at least half of the rooms are occupied on a given day? b. What is the probability that 100 or more rooms are occupied on a given day? c. What is the probability that 80 or fewer rooms are occupied on a given day?

A Myrtle Beach resort hotel has 120 rooms. In the spring months, hotel room occupancy is approximately $75 \%$
a. What is the probability that at least half of the rooms are occupied on a given day?
b. What is the probability that 100 or more rooms are occupied on a given day?
c. What is the probability that 80 or fewer rooms are occupied on a given day?

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials with the same probability of success. It is especially useful when there are two outcomes (such as ‘occupied’ or ‘not occupied’), and each trial (or room) is independent of the others.

Mean and Variance of the Binomial Distribution

For a binomial random variable with n trials and success probability p, the mean is given by np and the variance by np(1-p). These parameters provide insight into the expected number of successes and the spread of the distribution, which can be useful for approximations and understanding the pattern of occupancy.

Cumulative Probability

Cumulative probability refers to the probability that the random variable takes a value less than or equal to (or greater than or equal to) a certain threshold. In problems like these, calculations often involve summing binomial probabilities for a range of values or using complements to simplify the process.

Normal Approximation

When the number of trials is large and the probability of success is not too close to 0 or 1, the binomial distribution can be approximated using a normal distribution. This simplifies the process of finding cumulative probabilities, as the normal distribution has well-tabulated probabilities.

Complement Rule

The complement rule is a fundamental principle in probability stating that the probability of an event occurring is one minus the probability of it not occurring. This rule is often used to convert 'at least' scenarios into 'less than' scenarios, making it easier to calculate cumulative probabilities.

Transcript

00:02 We derive this data from the introduction of the problem.

00:05 We have a sample space of 120 rooms and a p or a success rate of 0 .75.

00:13 For the first question, we are to find the probability that at least half of the rooms are occupied on any given day.

00:23 And that is equal to the probability that x is greater than half of the rooms that is 120 divided by 2.

00:33 So we have x to be greater than or equal to 60.

00:39 This problem follows a binomial distribution, but since the n is very large, i'm going to approximate it using the normal distribution.

00:50 Therefore, we need to find the mean and the standard deviation.

00:58 Now, the mean is giving us n times p.

01:02 Therefore, we have 120 times 0 .75, and this is equal to 90.

01:11 The standard deviation is equal to square root of n times p multiplied by 1 minus p.

01:29 So we have 120 times 0 .75 multiplied by 1 minus 0 .75.

01:44 We have the square root of 22 .5 which is equal to 4 .74.

01:59 Now now we have our mean and standard deviation.

02:03 Head to use a normal approximation to the binomial distribution to compute the probability.

02:14 To do that, we need to find the z core corresponding to this value.

02:20 The formula is given as z is equal to x minus the mean divided by standard deviation.

02:26 So we have the probability that z is greater than or equal to the x value which is 60 minus the mean 90 divided by the standard deviation 4 .74.

02:52 So i forgot to include this.

02:55 When we are approximating from the binomial distribution to the normal distribution, since it is from a discrete to a continuous probability, we use what we call continuity and by continuity 60 becomes 60 minus 0 .5.

03:29 Which is equal to 59 .5...

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