91Ó°ÊÓ

According to Krantz (1992, p. 161), the probability of being injured by lightning in any given year is \(1 / 685,000 .\) Assume that the probability remains the same from year to year and that avoiding a strike in one year doesn't change your probability in the next year. a. What is the probability that someone who lives 80 years will never be struck by lightning? You do not need to compute the answer, but write down how it would be computed. b. According to Krantz, the probability of being injured by lightning over the average lifetime is \(1 / 9100 .\) Show how that probability should relate to your answer in part (a), assuming that average lifetime is about 80 years. c. Do the probabilities given in this exercise apply specifically to you? Explain. d. Over 300 million people live in the United States. In a typical year, assuming Krantz's figure is accurate, about how many people out of 300 million would be expected to be struck by lightning?

Step by step solution

01

Understanding the annual probability

According to the problem, the probability of being injured by lightning in any given year is \( \frac{1}{685,000} \). This probability remains constant each year.

02

Probability of never being struck over 80 years

The probability of not being struck by lightning in a given year is \( 1 - \frac{1}{685,000} \). For someone living 80 years, the probability of never being struck is the product of surviving each of these years without a strike, therefore:\[P(\text{never struck in 80 years}) = \left(1 - \frac{1}{685,000}\right)^{80}\]

03

Relating lifetime probability with part (a)

Krantz gives the probability of being injured over a lifetime of about 80 years as \( \frac{1}{9100} \). This implies that the probability of being struck at least once in 80 years is \( 1 - P(\text{never struck in 80 years}) = \frac{1}{9100} \).The formula established in step 2 should closely approximate this figure if the yearly probability holds true over the average lifetime.

04

Explanation of direct applicability

The probabilities describe an average across the population and might not directly apply to an individual due to varying personal circumstances such as geographical location and lifestyle. For example, someone who lives in a region with higher lightning activity might have a different probability.

05

Number of expected lightning injuries per year

Given the probability of \( \frac{1}{685,000} \), we can calculate the expected number of people struck per year in a population of 300 million as follows:\[\text{Expected per year} = 300,000,000 \times \frac{1}{685,000} = \frac{300,000,000}{685,000} \approx 438\] So, approximately 438 people would be expected to be struck by lightning each year in the U.S.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability of events

Probability is a way to measure how likely it is that something will happen. Probabilities are usually expressed as numbers between 0 and 1, where 0 means the event never happens, and 1 means it always happens.
For example, if you have a 1 in 10 chance of winning a game, the probability is 0.1 (or 10%). When you're considering events over time, like being struck by lightning, probabilities can be more complicated.
In this context, we define the probability of being injured by lightning in any given year as \( \frac{1}{685,000} \), which is indeed a very tiny chance.

• If the same event happens repeatedly, like surviving a year without being struck by lightning, probabilities must reflect each instance.
• Calculating the probability of an event occurring over multiple instances involves mathematical calculations.

In our case, it implies assuming that each year is independent, with its own probability that does not influence the next year's likelihood.

Lifetime risk

Lifetime risk refers to the probability that an event will happen to a person at least once during their lifetime. It's particularly useful in contexts like health where events are rare in any single year but could happen over a longer timeframe.
When discussing being struck by lightning, the lifetime probability is expressed as \( \frac{1}{9100} \). This number is significantly larger than the yearly risk because it's calculated to reflect the possibility over many years.
The lifetime probability is related to the yearly probability by acknowledging how small annual risks add up. After 80 years, the chance that at least one strike occurs becomes tangible.

• It assumes the individual will live for the average lifespan of 80 years.
• This calculation helps bridge understanding between annual and total lifetime risks.

Therefore, it gives a comprehensive picture of the risk faced by any one person over their lifetime.

Expected value

Expected value is a concept used to predict the average outcome of an event over many occurrences. It sums up all possible outcomes, each multiplied by the probability of that outcome.
In the context of lightning strikes, the expected value concept helps estimate how many people might be struck by lightning in a large population.
Given the probability of \( \frac{1}{685,000} \), you can calculate the expected number of lightning strikes among 300 million people in the U.S.: \[\text{Expected per year} = 300,000,000 \times \frac{1}{685,000} \approx 438\] This means around 438 people are statistically expected to be struck by lightning each year in such a large population.

• Expected value provides a way to see the bigger picture of rare events in large groups.
• While individual odds are tiny, they help aggregate into meaningful numbers at a community or national level.

This method is valuable for creating prevention and safety plans based on statistical averages.

Probability calculations

Doing probability calculations is about using known probabilities to find unknown ones.
When calculating the probability of not being struck by lightning over 80 years, you use the formula:\[ P(\text{never struck in 80 years}) = \left(1 - \frac{1}{685,000}\right)^{80} \]This formula takes the annual survival odds and multiplies them over 80 years.
The full calculation gives a very close approximation of the probability of not being struck even once. In part b of the exercise, we relate this to the given lifetime risk.

• This means using basic math principles and probability rules to do more intricate calculations.
• Such steps make complex predictions and assessments more accessible.

Understanding these calculations helps visualize how probability works over multiple events or years.