Question

At Perry’s Pumpkin Patch there are 50 pumpkins to choose from in four different colors. Five pumpkins are yellow, 12 are green, 6 are white and the rest are orange pumpkins. Answer the following questions. PLEASE SHOW STEPS/CALCULATIONS

d) Over on another field there are 300 pumpkins, 20 of which are white. You are going to randomly choose 10 pumpkins from this larger field. What is the probability that you have chosen at most one white pumpkin from the second field?

e) Is there an approximation that can be used on the second field to find the probability that you have chosen at most one white pumpkin? Justify why you can or cannot use an approximation and state its distribution, parameter(s), and support.

f) Find the approximate probability that you have chosen at most one white pumpkin from the second field.

Answer 1

(d) Here there are 300 pumpkins

WHite ones = 20

We have to choose = 10 pumpkins

If x is the number of white pumpkins out of 10.

so Here

Pr(x < =1 ; 10 ; 20 ; 200) = p(X =0) + p(X = 1)

= ²⁰C₀²⁸⁰C₁₀/ ³⁰⁰C₁₀ + ²⁰C₁²⁸⁰C₉/³⁰⁰C₁₀

= 0.4961 + 0.3661 = 0.8622

(e) Here we can approximate it with binomial distribution where n = 10 and p = 20/300 = 1/15

so here as sample proportion is less thaa 10% of the population size so we can approximate the distribution. Now the distribution is binomial, parameter(s) are n = 10 and p = 1/15 and support is x from 0 to 10.

(f) Pr(x < = 1; 10 ; 1/15) = p(x = 0) + p(x = 1)

= ¹⁰C₀ (1/15)⁰(14/15)¹⁰ + ¹⁰C₁ (1/15)¹(14/15)⁹

= 0.5016 + 0.3583

= 0.8600