Try every question on paper first, and keep fractions exact. The questions use invented situations and follow the skills in combined-event probability.
For a timed run, use the timed original practice session builder.
Questions
Q1. A fair coin is tossed and a fair four-sided die numbered 1 to 4 is rolled. List the sample space and find P(head and a number greater than 2).
Answer
H1, H2, H3, H4, T1, T2, T3, T4. That is 2 × 4 = 8 outcomes.
The outcomes with a head and a number greater than 2 are H3 and H4.
P = 2/8 = 1/4.
Q2. Two fair dice are rolled. Find the probability that the sum is 8.
Answer
The pairs are (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2). That is 5 outcomes out of 36.
P(sum 8) = 5/36.
Q3. State whether each pair is independent or dependent: (a) tossing a coin and rolling a die, (b) drawing two cards from a hand of five without replacement.
Answer
(a) Independent. The coin does not change the die.
(b) Dependent. After the first card is removed, four cards remain, so the second probability changes.
Q4. A box has 5 blue pens and 3 black pens. Two pens are drawn with replacement. Find P(both black).
Answer
P = 3/8 × 3/8 = 9/64.
Q5. Using the same box, the two pens are drawn without replacement. Find P(both black).
Answer
P = 3/8 × 2/7 = 6/56 = 3/28.
After one black pen is kept, 2 black pens remain among 7.
Q6. A bag has 4 red and 6 green counters. Two are drawn without replacement. Find the probability that one is red and one is green.
Answer
P(red then green) = 4/10 × 6/9 = 24/90. P(green then red) = 6/10 × 4/9 = 24/90.
Add both paths: 24/90 + 24/90 = 48/90 = 8/15.
Q7. A number is chosen at random from 1 to 12. Find P(a multiple of 3 or an even number).
Answer
Multiples of 3: {3, 6, 9, 12}, which is 4. Even numbers: {2, 4, 6, 8, 10, 12}, which is 6. Both: {6, 12}, which is 2.
P = (4 + 6 − 2)/12 = 8/12 = 2/3.
Q8. P(A) = 0.6, P(B) = 0.5, and A and B are independent. Find P(A and B) and P(A or B).
Answer
P(A and B) = 0.6 × 0.5 = 0.3.
P(A or B) = 0.6 + 0.5 − 0.3 = 0.8.
If you got these wrong
Listing slips (Q1, Q2) go back to drawing sample spaces systematically. Slips in Q3 to Q5 go to distinguishing independent and dependent events.
Tree slips (Q6) go to using tree diagrams with and without replacement. Overlap slips (Q7, Q8) go to combining mutually exclusive and non-exclusive events.
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