X is 1 for bike delivery and 0 for van delivery. Y is 1 for on time and 0 for late. What event does P(X=1,Y=0) describe?

• The parcel arrived on time, by either delivery method.
• The parcel came by bike, regardless of its arrival time.
• The parcel came by bike and arrived late.
• The parcel came by van and arrived on time.

Choose one of 24 users uniformly. App users: 9 subscribed, 3 not subscribed. Web users: 6 subscribed, 6 not subscribed. What is the marginal probability of being subscribed?

• 0.375
• 0.625
• 0.500

For events A and B with positive probabilities, P(A given B) must equal P(B given A).

Choose one of 25 parcels uniformly. Bike deliveries: 9 on time, 6 late. Van deliveries: 7 on time, 3 late. What is the joint probability that the parcel came by bike and was late? Give a decimal.

In a complete 3-by-3 joint table, one entry equals its row marginal multiplied by its column marginal. This alone proves that the variables are independent.

The complete joint table has X=1,2,3 and Y=0,1. In that Y order, the rows are X=1: 0.10,0.15; X=2: 0.20,0.15; X=3: 0.05,0.35. What is P(X at least 2)? Give a decimal.

P(B)>0. Within B, X takes only 1, 2 and 3. P(X=1 given B)=0.20 and P(X=2 given B)=0.30. What is P(X=3 given B)? Give a decimal.

The complete joint table has X=0,1 and Y=0,1,2. In that Y order, the rows are X=0: 0.10,0.15,0.05; X=1: 0.20,0.30,0.20. What is P(X=Y)? Give a decimal.

X and Y each take only 0 and 1. Their complete joint table is P(0,0)=0.12, P(0,1)=0.28, P(1,0)=0.18, P(1,1)=0.42, where P(x,y)=P(X=x,Y=y). Are they independent?

• No: unequal joint entries prove that the variables are dependent.
• Yes: all four entries equal their marginal products.
• Cannot tell: a complete joint table cannot establish independence.

P(A)=0.45 and P(B given A)=0.60. What is P(A and B)? Give a decimal.