MATH 225N Week 4 Additional Practice Questions: Probability & Discrete Distribution

, MATH 225N Week 4 Additional Practice Questions: Probability & Discrete
Distributions
Question 11: Union Probability
In a class of 40 students:
• 18 students play basketball
• 22 students play soccer
• 7 students play both sports
If a student is selected at random, what is the probability they play basketball
or soccer?
Answer:
Using the addition rule: P(A∪B)=P(A)+P(B)−P(A∩B)P(A∪B)=P(A)+P(B)−P(A∩B)
P(Basketball)=1840=0.45P(Basketball)=4018
=0.45P(Soccer)=2240=0.55P(Soccer)=4022
=0.55P(Both)=740=0.175P(Both)=407
=0.175P(Basketball or Soccer)=0.45+0.55−0.175=0.825P(Basketball or Soccer)=
0.45+0.55−0.175=0.825
Alternatively, from counts: 18+22−7=3318+22−7=33 students play at least one
sport.
P=3340=0.825P=4033=0.825

Question 12: Binomial - Probability of at most
A quality control inspector checks 12 items from an assembly line. Each item
has an 8% probability of being defective, independently. What is the
probability that at most 2 items are defective?
Answer:
• n=12n=12, p=0.08p=0.08, q=0.92q=0.92
• "At most 2" means X=0,1,or 2X=0,1,or 2
P(X=0)=(120)(0.08)0(0.92)12≈1×1×0.3677=0.3677P(X=0)=(012
)(0.08)0(0.92)12≈1×1×0.3677=0.3677P(X=1)=(121)(0.08)1(0.92)11≈12×0.08×0.3
996=0.3836P(X=1)=(112
)(0.08)1(0.92)11≈12×0.08×0.3996=0.3836P(X=2)=(122)(0.08)2(0.92)10≈66×0.00
64×0.4344=0.1835P(X=2)=(212
)(0.08)2(0.92)10≈66×0.0064×0.4344=0.1835P(X≤2)≈0.3677+0.3836+0.1835=0.9
348P(X≤2)≈0.3677+0.3836+0.1835=0.9348

Question 13: Expected Value in a Game