Probability theory and statistics

In summary, the conversation discusses the probability of eight different runners, whose times are independent after 100 minutes, scoring or not scoring. The solution involves using the maximum and minimum times of the runners to calculate the overall probability. The correct probabilities for all runners scoring and none scoring are (11/36)^8 and (25/36)^8 respectively. The incorrect approach used the probability of the runner's times falling between 100 and 125 minutes, which resulted in an incorrect probability. The use of max and min times is necessary in this case to accurately calculate the overall probability.
  • #1
Pouyan
103
8

Homework Statement


The time (minute) that it takes for a terrain runner to get around a runway is a random variable X with the tightness function
fX = (125-x)/450 , 95≤x≤125

How big is the probability of eight different runners, whose times are independent after 100 minutes:

a) Everyone has scored?
b) nobody has scored?

Homework Equations


I do know the distribution function is :

FX(t) = P(X≤t) = ∫ (125-x)/450 * dx for x between 95 and t and we have
(250t - t2 -14725)/900

The Attempt at a Solution



The right solution is :
[/B]
n=8 and from X1 to Xn, we can describe independent stochastic variables, the time when the last one came into target:
Z=max(X1...Xn)
And the time the first hit the finish line:
Y= min(X1...Xn)
Further:
Fz(t) = P(Z<t)= P(max(X1,...Xn)<t)= FX1(t)...FXn(t)

in the same way:
FY(t)=1-(1-Fx1(t))...(1-FXn(t))
All in goal after 100 minutes: Z≤100
P(Z<100) = [FX(100)]8 = (11/36) 8

None in goal after 100 minutes: Y> 100
P(Y>100) = 1-FY(100)= (1-Fx(100))8 = (25/36)8

How do I think :
I interpreted "The time everyone has scored after 100 minutes" as P (X> 100)
And tried with P(100<X<125) = 1-Fx(100) = 0.69
Then 0.698 = 0.051
I know it's wrong, but this sentence "After 100 minutes ..." made me dizzy!

Why do we have to use max and min in this case?!
 
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  • #2
The two approaches look identical to me and the answers agree apart from rounding errors.
 

1. What is the difference between probability theory and statistics?

Probability theory is a branch of mathematics that deals with the study of random events and their likelihood of occurring. It provides a framework for understanding and predicting the outcomes of uncertain events. On the other hand, statistics is the science of collecting, analyzing, and interpreting data in order to make informed decisions and predictions. While probability theory focuses on the theoretical aspects of randomness, statistics applies these concepts to real-world data.

2. How is probability used in statistics?

In statistics, probability is used to quantify the uncertainty associated with data and to make predictions about future events. It allows us to calculate the likelihood of different outcomes occurring and to make informed decisions based on this information. Probability is also used to assess the reliability of statistical results and to determine the significance of relationships between variables.

3. What is the difference between descriptive and inferential statistics?

Descriptive statistics involves summarizing and describing a set of data, usually through measures such as mean, median, and standard deviation. It provides a way to organize and make sense of data, but does not make inferences about a larger population. On the other hand, inferential statistics uses sample data to make predictions or generalizations about a larger population. It involves applying probability theory to determine the likelihood of certain outcomes and to test hypotheses.

4. What is the importance of understanding probability and statistics?

Understanding probability and statistics is crucial in many fields, including science, medicine, economics, and engineering. It allows us to make sense of data and to draw meaningful conclusions from it. In addition, it helps us to make informed decisions based on evidence and to evaluate the reliability of research findings. Probability and statistics are also essential for making predictions and for understanding and managing risk.

5. What are some common applications of probability and statistics?

Probability and statistics have numerous applications in various fields. In medicine, they are used to assess the effectiveness of treatments and to analyze risk factors for diseases. In finance, they are used for risk management and to make investment decisions. In engineering, they are used to design experiments and to analyze data from experiments. In marketing, they are used to analyze consumer behavior and to make predictions about sales. Probability and statistics are also used in sports to analyze player performance and to make predictions about game outcomes.

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