Concept explainers
Expand Your Knowledge: Geometric Distribution; Agriculture
Approximately
(a) Write out a formula for the
(b) Find the probabilities that
(c) Find the probability that
(d) What is the expected number of apples that must be examined to find the first one with bitter pit? Hint: Use
Hint: See Problem 23.
(a)
The formula for the probability distribution of the random variable n.
Answer to Problem 24P
Solution: The required formula is
Explanation of Solution
Given: 3.6% of the Jonathan has bitter pit and n is a random variable, which shows that first Jonathan apple selected at random has bitter pit. So, the provided value is, p=0.036.
Calculation:
The random variable, n follows the geometric distribution with the probability of success,
The formula to calculate probability in geometric distribution is:
So, the formula of the probability distribution of the first apple selected has bitter pit can be found by using the above formula:
Thus, the formula of the probability distribution of the random variable n is
(b)
To find: The probability values for n=3, n=5 and n=12.
Answer to Problem 24P
Solution: The required values of probabilities are 0.03345, 0.0311 and 0.0241 respectively.
Explanation of Solution
Given: The provided values are p= 0.036 and n=3, 5 and 12.
Calculation:
The formula to calculate probability in geometric distribution is:
So, the probability that third Jonathan apple selected at random is the first one to have bitter pit can be calculated by using the geometric experiment formula:
The probability that fifth Jonathan apple selected at random is the first one to have bitter pit can be calculated by using the geometric experiment formula:
The probability that twelfth Jonathan apple selected at random is the first one to have bitter pit can be calculated by using the geometric experiment formula:
Interpretation: There is about 3.35%, 3.11% and 2.41% chance that
(c)
The probability that
Answer to Problem 24P
Solution: The estimated value of required probability is 0.8636.
Explanation of Solution
Given: The provided value is,
Calculation: The probability that
Now,
Interpretation: There is 86.36% chance that
(d)
The expected number of apples that are to be examined to find first one with bitter pit.
Answer to Problem 24P
Solution: The expected value is approximately 28.
Explanation of Solution
Given: 3.6% of the Jonathan has bitter pit. So, the provided value is,
Calculation:
The formula to calculate the expected value in geometric distribution is:
Now, substitute the provided value in the above formula as follows:
Therefore, the expected value is 28.
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Chapter 6 Solutions
Understanding Basic Statistics
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