Annual income under different demand patterns High $306,600 0.2 $153,300 0.3 $76,650 0.5 $317,550 0.3 $158,775 0.4 $79,400 0.3 $292,000 0.4 $146,000 0.4 $73,000 0.2 $319,375 0.5 $159,700 0.4 $79,850 0.1 Alternatives Average Low Option A Option B Option C Option D (a) Draw the decision tree for the entrepreneur. (b) Compute the expected income for each option. Which option has the highest expected income? (c) Suppose option A costs $82,000, option B costs $96,000, option C costs $118,000 and option D costs $145,000, and these costs have to be repaid within 1 year. Calculate the expected profit (income minus cost) of each option for the first year. Which option has the highest expected profit?
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
An entrepreneur is deciding how to subdivide a large old home to
convert it to a Bed-and-Breakfast (B&B). There are four alternatives: Option
A renovates all rooms, but leaves all walls intact, so the result would be two
double rooms and three single rooms on the top floor, and on the main floor two further single rooms with a shared bathroom. Option B would renovate only the second floor but combine the single rooms in the main floor to create a single room with a bathroom. Option C combines the main floor rooms to create a single room with a bathroom, and combines the three single rooms on the top floor to create two double rooms. Finally option D would create five double rooms, each with its own bathroom. There are three event possibilities, a high demand, an average demand, and a low demand. The entrepreneur believes the probability of each event depends on the layout and availability of the rooms. Below are the details of the predicted income and the probability of occurrence that accompany each option and demand situation:
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