Marbles from a vase
A vase holds 5 red and 4 blue marbles. You draw two of them without looking, one after the other, and you do not put the first one back. What is the chance that both are red? The first is red with chance 5/9. Then 8 marbles are left, 4 of them red, so the second is red with chance 4/8. Together: 5/9 × 4/8 = 20/72 = 5/18, about 27.8%. Click the vase to draw a marble without looking. Without replacement, every marble you take out changes what is left, so the chance of the next colour changes too. The tree diagram shows every way the draws can go: multiply the chances along a branch, and add up the branches that answer the question. Press Draw automatically to draw 1,000 rounds and watch the share of yes settle at the chance from the tree.
How it works
- Click the vase: a marble comes out and lands in the first dish, and the tree diagram lights up its branch.
- Click again for the second marble, and read how many of each colour were left and what the chance of red was then.
- Press Draw automatically to draw 1,000 rounds and watch the share of yes settle at 5/18 ≈ 27.8%.
Without replacement
Every marble you take out stays out. The vase gets smaller and its mix changes, so the chance of the second marble depends on the first. After a red one, 4 of the 8 marbles left are red: 4/8. After a blue one, 5 of the 8 are red: 5/8. The draws are dependent, and the chance on each branch of the tree diagram is a conditional chance: the chance of this colour, given the colours already drawn.
Multiply along the branches
The chance of one path through the tree is the product of the chances on its branches: red and then red is 5/9 × 4/8 = 20/72. That is the multiplication rule: P(A and B) = P(A) × P(B given A). The four paths red-red, red-blue, blue-red and blue-blue have chances 20/72, 20/72, 20/72 and 12/72, and together they make 72/72 = 1, because one of them always happens.
Add up the paths that count
A question such as "are both marbles the same colour?" is answered by more than one path: red-red and blue-blue. Their chances are added: 20/72 + 12/72 = 32/72 = 4/9. When most paths count, it is quicker to subtract the ones that do not: at least one red is 1 − 12/72 = 60/72 = 5/6. The app shows the calculation for every question.
With replacement
Put each marble back and the vase is the same for every draw: both red is then 5/9 × 5/9 = 25/81 ≈ 30.9%, a little more than 27.8%. The draws are independent. The bigger the vase, the smaller the difference: with 10 red and 10 blue, two red ones is 9/38 ≈ 23.7% without replacement and 25% with it. In a small vase of 2 red and 2 blue the difference is large: two of the same colour is 1/3 without replacement and 1/2 with it.
Frequently asked questions
What is the chance of drawing two red marbles from 5 red and 4 blue?
Without putting the first one back: 5/9 × 4/8 = 20/72 = 5/18, about 27.8%. With putting it back: 5/9 × 5/9 = 25/81, about 30.9%.
What is the difference between with and without replacement?
With replacement every draw starts from the same vase, so the draws are independent and the chances stay the same. Without replacement the vase changes after every draw, so each chance depends on what was drawn before.
Why do you multiply the chances in a tree diagram?
Because the second chance is a share of the first. Of all rounds, 5/9 start with a red marble, and of those, 4/8 go on with a second red one: half of five ninths is 5/18.
Does the order of the draws matter?
Not for the chance of a combination. Red then blue and blue then red both have chance 20/72 (5/9 × 4/8 and 4/9 × 5/8): the same numbers, multiplied in a different order. Exactly one red is the sum of both paths: 40/72 = 5/9.
Does this upload anything?
No. Everything is calculated and drawn on a canvas in your browser. Save downloads a PNG. Edit opens that image in the PictorX editor.
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