Question

An article in the New York Times reported that people who suffer cardiac arrest in New York City have only a 1 in 100 chance of survival. Using probability notation, an equivalent statement would be \(P(\) survival \()=0.01\) for people who suffer cardiac arrest in New York City. (The article attributed this poor survival rate to factors common in large cities: traffic congestion and difficulty finding victims in large buildings. Similar studies in smaller cities showed higher survival rates.) a. Give a relative frequency interpretation of the given probability. b. The basis for the New York Times article was a research study of 2,329 consecutive cardiac arrests in New York City. To justify the " 1 in 100 chance of survival" statement, how many of the 2,329 cardiac arrest sufferers do you think survived? Explain.

Short answer

The relative frequency interpretation of the given probability suggests that in a very large number of cardiac arrests in New York City, we would expect about 1% of them to survive. When justified with the data from the study, it suggests that from the 2,329 cardiac arrests, around 23 people would have survived.

Step-by-step solution

Understanding Relative Frequency Interpretation

The relative frequency interpretation of a probability refers to an estimate of the chance that a particular event will occur. It is based on observed data and calculated as the ratio of the number of observed occurrences of the event to the total number of trials. In this problem, a relative frequency interpretation of the given probability, \(P(\) survival \()=0.01\) , suggests that if we were to observe a very large number of cardiac arrests in New York City, we would expect about 1% of them to survive.

Calculation of The Number of Survivors

Next, to justify the 1 in 100 chance of survival statement, and figure out the number of survivors out of the 2,329 cardiac arrest sufferers, multiply the total number of sufferers, which is 2,329, by the survival probability of 0.01. This accounts for the given probability.

Justification and Explanation

The statement that 1 in 100 people survive cardiac arrest in New York City is justified by the fact that, of the 2,329 consecutive cardiac arrests in the city, approximately 2,329 * 0.01 = 23 people are expected to have survived. The numbers are provided by a study, justifying the relative frequency interpretation of the survival probability.

Key concepts

Relative Frequency Interpretation of Probability

When we talk about the relative frequency interpretation of probability, we are referring to how often an event occurs relative to the number of trials or opportunities for it to happen. It's like taking a tally of how many times something has happened and relating it to the total count of attempts or observations.

Consider a baseball player with a batting average of .300. This means that, relative to frequency, the player hits safely in 30% of his at bats over a season. It's calculated by dividing the number of hits by the total at bats. Similarly, the reported probability of survival from cardiac arrest (\( P(\text{survival}) = 0.01 \) is a statistical estimate indicating that, historically, for every 100 cases of cardiac arrest in New York City, about one person survives.

The interpretation is essentially a prediction based on past data. If New York City continues to have the same conditions, then this probability tells us that the pattern of survival is expected to stay consistent - roughly one survivor for every 100 instances of cardiac arrest. The value of using relative frequency is that it provides a practical way to understand and communicate the likelihood of events, especially in high-stakes situations like emergency medical response.

Understanding Probability Notation

Probability notation is a shorthand way to express the likelihood of an event. It allows statisticians, mathematicians, and students alike to quickly understand and calculate the odds of something happening. In our cardiac arrest example, the probability notation \( P(\text{survival}) = 0.01 \) is read as 'the probability of survival equals 0.01', or in plain terms, there's a 1% chance of surviving.

This notation is essential because it provides a common language for discussing uncertain outcomes. It can be broken down into parts: the 'P' stands for 'probability', and the event of interest is enclosed in parentheses. The equal sign then links this to the numerical likelihood - in this case, 0.01.

Being comfortable with reading and writing probabilities in this way is crucial for understanding studies, reports, and analyses across a variety of fields, not just survival statistics. It scales from everyday probabilities (like the chance of rain) to those with serious societal implications (like survival rates for illnesses).

Survival Rate Statistics

Survival rate statistics are vital numbers used in evaluating the effectiveness of healthcare systems, and understanding the risks associated with various conditions. These rates tell us the proportion of people who continue to live for a certain period of time after being diagnosed with a disease or experiencing a critical health event like a cardiac arrest.

In the context of our exercise, the survival rate for cardiac arrest in New York City is given as 1 in 100, derived from a study of 2,329 cases. If we were to convert this into a percentage, it shows a survival rate of 1%. To find the number of individuals who survived from the study, we would use the following calculation: \( 2,329 \times 0.01 = 23.29 \) - which, when rounded, suggests approximately 23 survivors.

Understanding these statistics helps public health officials to identify areas for improvement, such as faster emergency response times. It can also provide the public with valuable information on the likelihood of recovery from certain incidents, aiding in preventive measures and emergency preparedness.