Problem 74 - Basic Statistics for Business and Economics with Student

Question

Tim Bleckie is the owner of Bleckie Investment and Real Estate Company. The company recently purchased four tracts of land in Holly Farms Estates and six tracts in Newburg Woods. The tracts are all equally desirable and sell for about the same amount. a. What is the probability that the next two tracts sold will be in Newburg Woods? b. What is the probability that of the next four sold at least one will be in Holly Farms? c. Are these events independent or dependent?

Step-by-Step Solution

Answer

a. Probability is \(\frac{1}{3}\); b. Probability is \(\frac{13}{14}\); c. Events are dependent.

Step 1: Identify Total Tracts

First, we need to determine the total number of tracts available. Tim Bleckie's company owns 4 tracts in Holly Farms Estates and 6 tracts in Newburg Woods.The total number of tracts is:\[4 + 6 = 10\]

Step 2: Calculate Probability for Part (a)

For part (a), we want the probability that the next two tracts sold will both be in Newburg Woods. The probability can be found by calculating the probability of both tracts being Newburg Woods in consecutive sales, without replacement.1. Probability that the first tract is from Newburg Woods: \[ \frac{6}{10} = 0.6 \]2. Probability that the second tract is also from Newburg Woods: \[ \frac{5}{9} \] 3. Therefore, the probability for both is:\[0.6 \times \frac{5}{9} = \frac{1}{3}\]

Step 3: Calculate Probability for Part (b)

For part (b), we need the probability that at least one out of the four tracts sold is from Holly Farms. This can be solved by using the complement rule: First, find the probability that none of the tracts are from Holly Farms (all from Newburg Woods) and subtract from 1.1. Probability that all 4 are from Newburg Woods: - First tract: \( \frac{6}{10} \) - Second tract: \( \frac{5}{9} \) - Third tract: \( \frac{4}{8} \) - Fourth tract: \( \frac{3}{7} \)2. Probability that all four tracts are from Newburg Woods: \[ \frac{6}{10} \times \frac{5}{9} \times \frac{4}{8} \times \frac{3}{7} = \frac{1}{14} \]3. Probability that at least one is from Holly Farms: \[ 1 - \frac{1}{14} = \frac{13}{14} \]

Step 4: Evaluate Independence

For part (c), consider if the events are independent. Two events are independent if the occurrence of one event does not affect the probability of the other. In the context of this problem, since selling a tract reduces the total number of tracts available, the events are dependent. Each sale affects the outcome of future sales.

Key Concepts

Dependent Events

In probability theory, events are considered _dependent_ when the outcome of one event influences or alters the likelihood of the outcome of another event. This is opposed to independent events, where one event does not affect the probability of the other occurring.
For the exercise involving Bleckie Investment and Real Estate Company, the sale of a tract of land alters the probabilities of everything that follows.

• When a Newburg Woods tract is sold, it decreases the number of available tracts.
• This reduction changes the probability for the next sale, making each sale dependent on the results of the previous one.

For instance, if you sell a tract from Newburg Woods, the chances of the next tract also being from Newburg Woods changes because there is one less tract available. This is a classic example of how dependent events work in probability.

Complement Rule

The complement rule is an important concept in probability which helps in calculating the probability of at least one event occurring. It is often the easier route when dealing with complex problems.
The rule states that the probability of an event happening is equal to 1 minus the probability of it not happening. For example, if you're looking for the probability that at least one of the next four tracts sold is from Holly Farms, you can use the complement rule:

• First, calculate the probability that none of the sold tracts come from Holly Farms. This means all tracts would come from Newburg Woods.
• Subtract this probability from 1. This result gives you the probability that at least one tract comes from Holly Farms.

Using this rule simplifies the problem because calculating no tracts from Holly Farms involves fewer steps than considering various combinations where at least one tract is from Holly Farms.

Probability Without Replacement

Understanding probability without replacement is crucial when you're dealing with scenarios where each event affects the next. This is common in situations like the exercise involving land tracts. When dealing with probability without replacement, once a tract is sold, it's no longer available for future outcomes, which changes the sample size.

• For instance, if you initially have 6 tracts from Newburg Woods and 4 from Holly Farms, the first sale affects the probabilities of the subsequent sale.
• If a tract from Newburg Woods is sold first, the remaining tracts are now 5 from Newburg Woods and still 4 from Holly Farms, altering the total tracts to 9 instead of the initial 10.

This change in the sample space makes calculating probabilities more complex, as it doesn't follow the typical independent event calculation. It reflects real-world scenarios where choices made affect future options, crucial for understanding practical probability applications.