Survival in a Lake

Solution & Explanation

To solve the problem of determining the probability of survival for a person who is 5.8 feet tall and unable to swim in a lake with an average depth of 5.6 feet and a standard deviation of 1 foot, we can model the depth of the lake as a normally distributed random variable.

Step-by-step Solution:

1. Assumptions:

   • The lake's depth follows a normal distribution, denoted as $D \sim N(\mu, \sigma^2)$, where $\mu = 5.6$ feet and $\sigma = 1$ foot.
   • The person will survive if the depth of the lake is less than or equal to their height, i.e., $D < 5.8$ feet.

2. Calculate the Z-score:

   • The Z-score is a measure of how many standard deviations an element is from the mean.
   • Formula for Z-score: $Z = \frac{X - \mu}{\sigma}$
   • Here, $X = 5.8$, $\mu = 5.6$, and $\sigma = 1$.
   • Substitute these values into the formula: $Z = \frac{5.8 - 5.6}{1} = 0.2$

3. Find the Probability:

   • We need to find $P(D < 5.8)$, which translates to finding $P(Z < 0.2)$ in the standard normal distribution.
   • Using a Z-table or standard normal distribution calculator, we find: $P(Z < 0.2) \approx 0.5793$
   • This means there is approximately a 57.93% chance that the depth of the lake will be less than 5.8 feet, and thus, the person will survive.

Explanation:

• Normal Distribution:

  • The assumption of a normal distribution is reasonable when dealing with natural phenomena like lake depths, especially when given a mean and standard deviation.

• Z-Score Interpretation:

  • The Z-score of 0.2 indicates that 5.8 feet is 0.2 standard deviations above the mean depth of the lake.

• Use of Cumulative Distribution Function (CDF):

  • The CDF provides the probability that a normally distributed random variable is less than or equal to a certain value, which in this case is the height of the person.

• Practical Considerations:

  • While the model provides a statistical probability, real-world factors such as variations in lake depth, swimming ability, and safety measures are not considered in this simplified model.

In conclusion, based on the normal distribution model, the probability of survival for the individual in this scenario is approximately 57.93%.