Math Homework Help!! Permutations, Combinations,Counting Techniques

Question

There are 10 puppies. 6 are male, 4 are female. If you pick 3 puppies at random, what is the probability they will all be the same gender?

Answer 1

You can answer this question in two different ways and get to the same answer 1/5 as follows:

 

1) Using dependent events and applying the multiplication rule with the conditional probability:

P(3males) = P(male and male and male) = 6/10*5/9*4/8 = 1/6

P(3females) = P(female and female and female) = 4/10*3/9*2/8 = 1/30

Total Probability = P(3males) or P(3females) = 1/6 + 1/30 = 6/30 = 1/5

 

2) Using Hyper Geometric Distribution:

P(3males) = ₆C_{3 4}C₀ / ₁₀C_{3 = 20/120 = 1/6}

P(3females) = ₆C_{0 4}C₃ / ₁₀C₃ = 4/120 = 1/30

and the total Probability, as before, is 1/5

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer 2

the total possible number of outcomes here can be modeled using combination of 10, choose 3

C(10,3) = 120

from there then we want to calculate the number of ways to pick three puppies that are all the same gender ... so that means computing the chances that all 3 are male or all 3 are female

Male: C(6,3) = 20

Female: C(4,3) = 4

meaning that the end probability for this scenario is

(20 + 4) / 120 = 0.2

so the probability of picking 3 puppies that are all the same gender is 0.2 or 20%