Problem 13 - Algebra 2 and Trigonometry

Question

In \(7-14,\) answers can be rounded to four decimal places. Using a special type of fishing hook, researchered that the hooking mortality rate was approximately 8\(\%\) . That is, of the fish caught and released with this hook, about 8\(\%\) eventually died. A fisherman caught and released 5 fish using this hook. What is the probability that all 5 fish will live?

Step-by-Step Solution

Answer

The probability that all 5 fish will live is 0.6591.

Step 1: Understanding the Problem

We need to find the probability that all 5 fish caught and released will survive. The mortality rate per fish is 8%, which means each fish has a 92% chance of surviving.

Step 2: Calculating Survival Probability for One Fish

If there is an 8% mortality rate, the probability of one fish surviving is \[ P( ext{survive}) = 1 - 0.08 = 0.92 \] Thus, each fish has a 0.92 probability of living.

Step 3: Applying the Probability for All Fish

Since the events are independent (the survival of one fish doesn't affect the others), the probability that all 5 fish survive is the product of their individual probabilities:\[ P( ext{all survive}) = 0.92^5 \]

Step 4: Performing the Calculation

Calculate the probability:\[ P( ext{all survive}) = 0.92^5 = 0.6591 \] This is the probability rounded to four decimal places.

Key Concepts

Mortality Rate

The concept of mortality rate plays a crucial role in understanding probabilities related to life and death events. In this context, the term refers to the proportion or percentage of individuals (such as fish in our exercise) that die after a certain event—in this case, being caught and released using a special type of fishing hook. Here, the mortality rate is given as 8%, which simply means that out of every 100 fish released, approximately 8 will die.
To comprehend this better, consider the mortality rate as a measure of risk. A lower mortality rate indicates a lower likelihood of death after the event, whereas a higher rate suggests a greater risk. Therefore, with an 8% mortality rate, the odds are relatively low, but still significant enough to consider when calculating probabilities of survival.

Independent Events

Independent events are critical when dealing with probabilities where more than one outcome is being considered. An event is independent if its occurrence does not affect the outcome of another event. In the fish mortality exercise, this means that each fish's survival is unaffected by whether the other fish survive or not.

In probability terms, for two events to be independent, the probability of both events occurring is the product of their individual probabilities. This concept simplifies computations significantly.
Given that the survival of one fish does not influence the others, we can multiply the survival probabilities of each fish to find the total probability that all survive. This principle of independence permits us to ascertain overall survival probabilities more easily and accurately.

Survival Probability

Understanding survival probability is about determining the likelihood of an entity avoiding a particular negative outcome, such as death. In our example, if the mortality rate is 8%, then the survival probability for each fish is 92%. This is calculated by subtracting the mortality rate from 100%, leading to the equation:
\[ P(\text{survive}) = 1 - 0.08 = 0.92 \]
Thus, each fish has a 92% chance of surviving after being caught.
Survival probabilities are helpful when predicting how certain variables will impact the longevity of a subject. It helps to balance the focus on positive outcomes against the potential risks, in this case, calculating how likely it is that all fish will survive.

Probability Calculation

Probability calculation is the process of determining the likelihood of a specific outcome. In the problem at hand, calculating the probability for multiple entities involves using the formula for independent events. Armed with the survival probability for a single fish, the probability that all 5 fish survive can be computed as follows:
\[ P(\text{all survive}) = 0.92^5 = 0.6591 \]
This formula results from the independence of fish survival, and represents the product of each individual survival probability raised to the number of events (in this case, the 5 fish).
Performing these calculations requires a basic understanding of exponentiation and multiplication of probabilities, which allows us to effectively predict the odds of combined independent events occurring.