KBAT Problems
KBAT Problem: A Two-Stage Probability
A KBAT problem where two events happen in sequence and a tree diagram organises the outcomes, the challenge is deciding whether the events are independent.
Independent or not?
Two balls are drawn from a bag. If the first is replaced, the second draw has the same probabilities; if not, they change.
Deciding "with or without replacement" before drawing the tree is the KBAT judgement that sets up everything else.
Multiply along, add across
On a tree diagram you multiply probabilities along each branch to a final outcome, then add the outcomes that satisfy the question. Showing the tree with probabilities on every branch earns the method marks.
Understand the problem
A bag holds 3 red and 2 blue marbles, five in all. Two marbles are drawn one after the other without replacement.
Find the probability of getting one of each colour. What it really asks: first decide that the draws are not independent, because not replacing the first marble changes the second draw's probabilities, that judgement sets up the whole tree.
Plan and solve
- Without replacement, after one marble is taken only 4 remain, so the second denominator is 4, not 5.
- Draw the tree. Path red then blue: 3/5 × 2/4 = 6/20 = 3/10.
- Path blue then red: 2/5 × 3/4 = 6/20 = 3/10.
- One of each is either path, so add them: 3/10 + 3/10 = 6/10 = 3/5.
Check and a variant
Check using the complement: P(both red) = 3/5 × 2/4 = 3/10 and P(both blue) = 2/5 × 1/4 = 1/10, so P(same colour) = 4/10 = 2/5, and one of each = 1 − 2/5 = 3/5, matching. A variant the examiner could add: draw with replacement instead.
Then each draw uses denominator 5, and P(one of each) = 2 × (3/5 × 2/5) = 12/25, a different answer, proving the independence decision matters.
Frequently asked questions
How do I decide whether events are independent?
Ask whether the first outcome changes the second one's probabilities. With replacement, the bag resets, so the draws are independent and both denominators stay the same.
Without replacement, the counts change, so they are dependent. Read the question for words like replaced or kept before you draw the tree, because everything else depends on it.
Why do I multiply along branches but add between paths?
You multiply along a branch because both stages must happen in sequence, and 'and' means multiply. You add between separate paths because they are different ways to get the same overall result, and 'or' means add.
Keeping the and-multiply, or-add rule straight is the core skill a tree diagram is testing.
Can I always check a probability answer with the complement?
Often, and it is a strong habit. If the outcomes split neatly into your event and its complement, computing the complement and subtracting from 1 gives an independent check.
Here same colour plus one of each must total 1, and they do. If your two routes to the answer disagree, you have found a slip to fix.