Probability I · Conditional probability and independence
Roll two fair six-sided dice independently and record the ordered pair
Problem
Roll two fair six-sided dice independently and record the ordered pair. Let A be the event that at least one die shows six, and let B be the event that the sum is at least nine. Find \(\Pr(A\mid B)\) and \(\Pr(B\mid A)\) by counting outcomes in the appropriate restricted sample spaces. Are A and B independent? Explain why reversing the condition changes the denominator even though the intersection is the same.
Hint
List ordered pairs with sums nine through twelve.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
The answer check and full solution for this problem come with ProofAnvil Practice membership ($19 USD monthly). See membership. Or start with the free Probability I sample problem: Try the free sample problem.
More Probability I practice problems
- A spinner has four equally likely sectors, labelled A₁,A₂,B,CProbability spaces, axioms, and rigorous counting
- Choose a three-element subset uniformly from {1,…,9}, and let M be its largest elementProbability spaces, axioms, and rigorous counting
- Independent rounds of a game have three possible results: win with probability 2/7…Conditional probability and independence
- Roll two fair six-sided dice independently and let X be the absolute difference between…Random variables and distribution functions
- Two independent attempts have success probabilities 1/5 and 4/5Random variables and distribution functions
- A random variable X takes values 0, 1, and 3 with probabilities 1/2,1/3,1/6, respectivelyExpectation, moments, and transforms
- A sorting device independently sends each of six incoming pieces to tray A, B, or C…Joint distributions and conditional expectation
- Conditioning changes the interval on a triangular supportJoint distributions and conditional expectation
- A weighted triangle is not uniform on its conditional slicesJoint distributions and conditional expectation