Chapter 6.3, Problem 24P

Expand Your Knowledge: Geometric Distribution; Agriculture

Approximately $3.6\%$ of all (untreated) Jonathan apples had bitter pit in a study conducted by the botanists Ratkowsky and Martin (Source: Australian Journal of Agricultural Research. Vol. 25. pp. 783-790). (Bitter pit is a disease of apples resulting in a soggy core, which can be caused either by overwatering the apple tree or by a calcium deficiency in the soil.) Let n be a random variable that represents the first Jonathan apple chosen at random that has bitter pit.

(a) Write out a formula for the probability distribution of the random variable n.

(b) Find the probabilities that $n=3.n=5.andn=12.$

(c) Find the probability that $n\ge 5.$

(d) What is the expected number of apples that must be examined to find the first one with bitter pit? Hint: Use $\mu$ for the geometric distribution and round.

Hint: See Problem 23.

To determine

The formula for the probability distribution of the random variable n.

Answer to Problem 24P

Solution: The required formula is $P\left(n\right)=\left(0.036\right){\left(0.964\right)}^{n-1}$

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, $p=0.036$.

The formula to calculate probability in geometric distribution is:

$P\left(n\right)=p{\left(1-p\right)}^{n-1}$

So, the formula of the probability distribution of the first apple selected has bitter pit can be found by using the above formula:

$\begin{array}{c}P\left(n\right)={0.036}^1\times {0.964}^{n-1}\\ =0.036\times {\left(0.964\right)}^{n-1}\end{array}$

Thus, the formula of the probability distribution of the random variable n is $P\left(n\right)=\left(0.036\right){\left(0.964\right)}^{n-1}$.

To determine

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:

$P\left(n\right)=p{\left(1-p\right)}^{n-1}$

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:

$\begin{array}{c}P\left(3\right)={0.036}^1\times {0.964}^{3-1}\\ =0.036\times {0.964}^2\\ =0.03345\end{array}$

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:

$\begin{array}{c}P\left(5\right)={0.036}^1\times {0.964}^{5-1}\\ =0.036\times {0.964}^4\\ =0.0311\end{array}$

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:

$\begin{array}{c}P\left(12\right)={0.036}^1\times {0.964}^{12-1}\\ =0.036\times {0.964}^{11}\\ =0.0241\end{array}$

Interpretation: There is about 3.35%, 3.11% and 2.41% chance that $n=3$, $n=5$ and $n=12$, respectively.

To determine

The probability that $n\ge 5$.

Answer to Problem 24P

Solution: The estimated value of required probability is 0.8636.

Explanation of Solution

Given: The provided value is, $p=0.036$.

Calculation: The probability that $n\ge 5$ can be calculated as:

$P\left(n\ge 5\right)=1-P\left(n=1\right)-P\left(n=2\right)-P\left(n=3\right)-P(n=4)$

Now,

$\begin{array}{c}P\left(n\ge 5\right)=1-\left[P\left(n=1\right)-P\left(n=2\right)-P\left(n=3\right)-P(n=4)\right]\\ =1-\left[{\left(0.036\right)}^1{\left(0.964\right)}^0-{\left(0.036\right)}^1{\left(0.964\right)}^1-{\left(0.036\right)}^1{\left(0.964\right)}^2-{\left(0.036\right)}^1{\left(0.964\right)}^3\right]\\ =0.8636\end{array}$

Interpretation: There is 86.36% chance that $n\ge 5$.

To determine

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, $p=0.036$.

Calculation:

The formula to calculate the expected value in geometric distribution is:

$\mu =\frac{1}{p}$

Now, substitute the provided value in the above formula as follows:

$\begin{array}{c}\mu =\frac{1}{0.036}\\ =27.78\\ \approx 28\end{array}$

Therefore, the expected value is 28.