Answered

Using the geometric probability distribution after 500 trials, determine the following probability. Fill in your answer in the blank space below.


  1. $P(X=1)=0.176$
  2. $P(X>1)=0.824$
  3. What is $P(X<6)$?




Number of Trials

500







$X$		$P(X)$

1		0.176

2		0.158

3		0.106

4		0.106

5		0.078

6		0.052

7		0.034

8		0.042

9		0.046

10		0.026







$X$		$P(X)$

11		0.022

12		0.014

13		0.032

14		0.025

15		0.018

16		0.018

17		0.006

18		0.008

19		0.004

20		0.002






$X$		$P(X)$

21		0.01

22		0.002

23		0.002

24		0.002

25		0.002

26		0.002

27		0

28		0

29		0

\ge 30		0.000

Verified Answer :

  • Identify the outcomes that satisfy \(X < 6\), which are \(X \in \{1, 2, 3, 4, 5\}\).
  • Extract the probabilities from the distribution table: \(P(1)=0.176, P(2)=0.158, P(3)=0.106, P(4)=0.106, P(5)=0.078\).
  • Sum these values to find the probability: \(P(X < 6) = 0.176 + 0.158 + 0.106 + 0.106 + 0.078 = 0.624\).
  • The final result is \(\boxed{0.624}\).

Explanation

  1. Analyze the problem and identify relevant events

We are working with a provided empirical probability distribution based on 500 trials. We need to find the probability \(P(X < 6)\). This probability is the sum of the probabilities of all discrete outcomes where \(X\) is less than 6. Since \(X\) takes positive integer values in this distribution, the outcomes we are looking for are \(X = 1, X = 2, X = 3, X = 4,\) and \(X = 5\).

  1. Extract the data from the probability distribution table

From the provided tables, we extract the individual probabilities for these values:

  • \(P(X=1) = 0.176\)
  • \(P(X=2) = 0.158\)
  • \(P(X=3) = 0.106\)
  • \(P(X=4) = 0.106\)
  • \(P(X=5) = 0.078\)
  1. Calculate the sum of probabilities

To calculate \(P(X < 6)\), we sum the probabilities identified in the previous step:
\[P(X < 6) = P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)\]
Substituting the numerical values:
\[P(X < 6) = 0.176 + 0.158 + 0.106 + 0.106 + 0.078\]
Calculating the sum:
\[0.176 + 0.158 = 0.334\]
\[0.334 + 0.106 = 0.440\]
\[0.440 + 0.106 = 0.546\]
\[0.546 + 0.078 = 0.624\]
Thus, the total probability is \(0.624\).

Examples

This type of analysis is common in quality control and data science. For instance, if you are analyzing the number of attempts a machine takes to successfully complete a process, this probability distribution helps predict the likelihood that the process succeeds within 5 attempts, allowing you to estimate reliability and performance efficiency in a real-world setting.

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