What is the probability that a child has seen:

Question

Buddy Vosburg, a child psychologist, was planning a study of response to certain aspects of the movies The Lion King, Shrek, and Finding Nemo. Upon surveying a group of 55 children, he determined the following: 17 had seen The Lion King 17 had seen Shrek 23 had seen Finding Nemo 6 had seen The Lion King and Shrek 8 had seen The Lion King and Finding Nemo 10 had seen Shrek and Finding Nemo 2 had seen all three of these movies

1. What is the probability that a child has seen:
a. Exactly two of these movies?
b. Exactly one of these movies?
c. None of these movies?
d. The Lion King but neither of the other movies?
e. At least Shrek and Finding Nemo

Answer 1

a.) 2 had seen all three, therefore subtract 2 from each of the groups that have seen at least two.

Therefore, 4 have seen only Lion King and Shrek, 6 have seen only Lion King and Finding Nemo, 8 have seen only Shrek and Finding Nemo. Add these up, 4 + 6 + 8 = 18, therefore in a group of 55 children the probably that a child has seen exactly two of these movies is 18/55 = 0.32 or 32%.

 

b.) Since 6 have seen Lion King and Shrek and 8 have seen Lion King and Finding Nemo, while subtracting the fact that 2 have seen all three means 8 + 6 - 2 = 12 have seen exactly two movies that include Lion king. Subtract 12 from the number that have seen Lion King leaves 17 - 12 = 5 have seen only Lion King. Continuing this for Shrek, we have 6 + 10 - 2 = 14 and therefore 17 - 14 = 3 have seen only Shrek. Lastly, for Finding Nemo, we have that 8 + 10 - 2 = 16 have seen only Finding Nemo and another movie, therefore 23 - 16 = 7 have seen only Finding Nemo. Add up these results, 5 + 3 + 7 = 15, therefore out of a group of 55 children the probability that a child has seen exactly one movie is 15/55 = 0.27 = 27%

 

c.) We are given that 2 children have seen all three movies. We have concluded that 18 have seen only two movies, and that 15 have only seen one movie. 2 + 18 + 15 = 35. Out of 55 children that leaves 55 - 35 = 20 children who have seen none of these movies, therefore the probability is 20/55 = 0.36 = 36%

 

d.) We concluded in 'b' which followed from 'a' that 5 children have seen only the Lion King and therefore 5/55 = 0.09 = 9%

 

e.) We are given that 10 children have seen Shrek and Finding Nemo, therefore 10/55 = 0.18 = 18%

 

We can check our answers for parts a through c, since the total should add up to the sample size. If we add the number of children that have seen all three (2), the number that have seen only two (18), the number that have seen only one (15) and the number that have seen none (20) we get 2 + 18 + 15 + 20 = 55.