Question

At the nursery, there were 6 potted plants that were supposed to grow red flowers and 8 potted plants that were supposed to grow green flowers. Freddie bought 4 flower pots but cannot for the life of him remember what colour they were supposed to be. He decided that when the plants bloom, for every red flower that grows, he will buy a scratch ticket. Assume his scratch tickets have a 25% chance of being winners and the chance of winning is independent. Let X be the number of red flowers that bloom and Y be the number of winning scratch tickets.

1. (A) [1 mark]
   Identify the distribution of X.

2. (B) [3 marks]
   Find the joint probability distribution of X and Y .

3. (C) [2 marks]
   Find P (X = Y ).

Answer 1

Here 6 potted plants that were supposed to grow red flowers and 8 potted plants that were supposed to grow green flowers.

Frddie bought 4 flowers.

. Let X be the number of red flowers that bloom and Y be the number of winning scratch tickets.

(a) Here the distribution of X is hypergeometric where the parameters are

N = 14, K = 6, n = 4

so here

p(x) = ⁶C_x⁸C_(4-x)/¹⁴C₄

p(0) = ⁶C₀⁸C₄/¹⁴C₄ = 0.0699

p(1) = ⁶C₁⁸C₃/¹⁴C₄ = 0.3357

p(2) = ⁶C₂⁸C₂/¹⁴C₄ = 0.4196

p(3) = ⁶C₃⁸C₁/¹⁴C₄ = 01598

p(4) = ⁶C₄⁸C₀/¹⁴C₄ = 0.0150

(b) Here for Y have the binomial distribution with parameter n = x and p = 0.25

so here

p(y) = ^xC_y (0.25)^y(0.75)^(4-y)

P(X,Y) =  (⁶C_x⁸C_(4-x)/¹⁴C₄ ) *  (⁶C_x⁸C_(4-x)/¹⁴C₄ ) ; 0 < = X < = 4 ; 0 < = y < = X

So now we will make a joint probability table

x/y	0	1	2	3	4	Total
0	0.0699	0	0	0	0	0.0699
1	0.251748	0.083916	0	0	0	0.3357
2	0.236014	0.157343	0.026224	0	0	0.4196
3	0.067433	0.067433	0.022478	0.002498	0	0.1598
4	0.004741	0.006322	0.003161	0.000702	0.0000585	0.0150

(c) p(x= Y) = p(0,0) + P(1,1) + P(2,2) + p(3,3) + p(4,4)

= 0.0699 + 0.083916 + 0.0262 + 0.002498 + 0.0000585 = 0.1826