# Probability Distributions, Long Questions (Question 4 & 5)

Question 4:
The diameter of oranges harvested from a fruit orchard has a normal distribution with a mean of 3.2 cm and a variance of 2.25 cm.
Calculate
(a) the probability that an orange chosen at random from this fruit orchard has a diameter of more than 3.8 cm.
(b) the value of k if 30.5 % of the oranges have diameter less than k cm.

Solution:
µ = 3.2 cm
σ2 = 2.25cm
σ = √2.25 = 1.5 cm
Let X represents the diameter of an orange.
X ~ N (3.2, 1.52)
(a)
$\begin{array}{l}P\left(X>3.8\right)\\ =P\left(Z>\frac{3.8-3.2}{1.5}\right)\\ =P\left(Z>0.4\right)\\ =0.3446\end{array}$

(b)

Question 5:
The masses of tomatoes in a farm are normally distributed with a mean of 130g and standard deviation of 16g. Tomato with weight more than 150g is classified as grade ‘A’.
(a) A tomato is chosen at random from the farm.
Find the probability that the tomato has a weight between 114g and 150g.
(b) It is found that 132 tomatoes in this farm are grade ‘A’.
Find the total number of tomatoes in the farm.

Solution:
µ = 130
σ = 16
(a)
$\begin{array}{l}P\left(1141\right)-P\left(Z>1.25\right)\\ =1-0.1587-0.1056\\ =0.7357\end{array}$
(b)
Probability of getting grade ‘A’ tomatoes,
P(X > 150) = P(Z > 1.25)
= 0.1056