204 / 205 - EXERCISES FOR SECTION 4.2
18- WHAT ARE EQUALY LIKELY OUTCOMES?
19- DESCRIBE THE DIFFERENCE BETWEEN AN IMPOSSIBLE EVENT AND A CERTAIN EVENT IN PROBABILITY. BE SURE TO PROVIDE EXAMPLES.
26- ALZHEIMER'S DISEASE. THE BAR GRAPH IN FIGURE 4.3 SHOWS THE PERCENTAGE OF PEOPLE 65 OR OLDER WHO HAVE BEEB DIAGNOSED WITH PROBALE ALZHEIMER'S DISEASE. FOR EXAMPLE THERE IS A 3% CHANCE THAT A PERSON WHO IS BETWEEN THE AGES OF 65 AND 74 HAS ALZHEIMER'S DISEASE. THUS, THE PROBABILITY IS 3/100 (SEE CHAPTER 12 FOR INFORMATION ON CONVERTING A PERCENT TO A FRACTION):
A. IF WE RANDOMLY SELECT AN 80YEAR OLD PERSON, WHAT IS THE PROBABILITY THAT THIS PERSON WILL HAVE ALZHEIMER'S DISEASE?
B. IF WE RANDOMLY SELECT A PERSON WHO IS AT LEAT 85 YEARS OLD, WHAT IS THE PROBABILITY THAT THIS PERSON DOES NOT HAVE ALZHEIMER'S DISEASE?
C. ARE THE PROBABILITIES SHOWN IN FIGURE 4.3 THEORETICAL OR EMPIRICAL?WHY?
FIGURE 4.3
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AGE FACTOR
PERCENTAGE OF PEOPLE 65
AND OLDER WITH PROBABLE
ALZHEIMER'S DISEASE
6-74 -----3%
75-84------- 19%
85-PLUS -------- 47%
SOURCE JOURNAL OF THE AMERICAN
MEDICAL ASSOCIATION
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18- Equally likely outcomes are outcomes in a probability experiment that have the same chance of occurring. When all outcomes are equally likely, each outcome has an equal probability of occurring. For example, when rolling a fair six-sided die, each of the numbers 1 to 6 is equally likely to come up.
19- In probability, an impossible event refers to an outcome that has no chance of occurring. The probability of this event is 0 because it cannot happen. For example, when flipping a fair coin, getting tails and heads at the same time is an impossible event.
On the other hand, a certain event is an outcome that is guaranteed to happen. The probability of this event is 1 because it is certain to occur. For example, when rolling a fair six-sided die, the probability of getting a number between 1 and 6 is a certain event since it will always happen.
26-
a. To find the probability that an 80-year-old person has Alzheimer's disease, we look at the information provided in Figure 4.3. It states that there is a 3% chance for people between the ages of 65 and 74 to have the disease. Since 80 falls within this age range, the probability remains the same. Therefore, the probability is 3/100 or 0.03.
b. To find the probability that a person who is at least 85 years old does not have Alzheimer's disease, we subtract the probability of having the disease from 1. According to Figure 4.3, the probability for people 85 and older to have the disease is 47%. So, the probability of not having the disease is 1 - 0.47 = 0.53 or 53%.
c. The probabilities shown in Figure 4.3 are theoretical probabilities. The probabilities are based on the given information and represent the calculated chance of each event occurring based on the age groups and diagnosis rates. The information provided is not the result of actual observations or experiments, which would make them empirical probabilities.
18- Equally likely outcomes refer to a situation where each possible outcome of an event has an equal chance of occurring. For example, when flipping a fair coin, there are two equally likely outcomes: heads or tails. Each outcome has a 50% chance of occurring.
19- In probability, an impossible event is one that has no chance of occurring. It has a probability of 0. For example, rolling a 7 on a fair six-sided die is an impossible event because the maximum number that can be rolled is 6. On the other hand, a certain event is one that is guaranteed to happen. It has a probability of 1. For example, if you draw a card from a standard deck, it is certain that you will draw a card, so the probability of drawing a card is 1.
26-
a. To find the probability that an 80-year-old person has Alzheimer's disease, we can refer to Figure 4.3. According to the graph, the percentage of people 65 or older with probable Alzheimer's disease is 19% for those aged 75-84. Since an 80-year-old person falls within this age range, we can say that the probability is 19/100 or 0.19.
b. To find the probability that a person who is at least 85 years old does not have Alzheimer's disease, we can subtract the probability of having the disease from 1. According to the graph, the percentage of people 65 or older with probable Alzheimer's disease is 47% for those aged 85 and above. Therefore, the probability of not having Alzheimer's disease for an 85+ year old person is 1 - 47/100 = 53/100 or 0.53.
c. The probabilities shown in Figure 4.3 are theoretical probabilities. They are based on mathematical calculations and assumptions using available data. The percentages represent the estimated likelihood of having Alzheimer's disease for different age groups.