Probability tree diagram maker

How to read a probability tree

Each stage is one thing that happens: a coin flip, a pick from a bag, a day's weather. From each point, one branch leaves for every outcome that could happen next, labelled with its probability. A path runs from the dot on the left to one end on the right and is one complete story, such as red, then blue.

• Multiply along a path. Red then blue is P(red first) × P(blue second, given red first).
• Add across paths. "One of each colour" happens on two paths, red-blue and blue-red, so you add their probabilities.
• Check the sums. The branches leaving any point add up to 1, and so do all the path ends. If a set of branches adds to 9/10, a branch is missing or a number is wrong; this page flags it and keeps your numbers as typed instead of rescaling them.

With and without replacement

With replacement, the item goes back before the next pick, so every stage looks the same: from 3 red and 2 blue, red is 3/5 every time. Without replacement, the bag shrinks by one after each pick. The denominator drops from 5 to 4, and the colour you just took loses one, so after a red the second pick is red 2/4 and blue 2/4, but after a blue it is red 3/4 and blue 1/4. The fractions are left unreduced on the branches (2/4, not 1/2) so you can see where they came from; each path's result is then reduced.

Worked example: two picks from 3 red and 2 blue, kept out

1. First pick: red 3/5, blue 2/5.
2. After a red there are 2 red and 2 blue left: red 2/4, blue 2/4. After a blue there are 3 red and 1 blue: red 3/4, blue 1/4.
3. Multiply each path: RR = 3/5 × 2/4 = 6/20 = 3/10; RB = 3/5 × 2/4 = 3/10; BR = 2/5 × 3/4 = 3/10; BB = 2/5 × 1/4 = 2/20 = 1/10.
4. Check: 3/10 + 3/10 + 3/10 + 1/10 = 1.
5. One of each colour: P(RB) + P(BR) = 3/10 + 3/10 = 3/5, or 0.6.

Put back instead, and RR becomes 3/5 × 3/5 = 9/25 (0.36), while one of each is 6/25 + 6/25 = 12/25 (0.48). Taking a red out makes a second red less likely, which is the whole difference.

How to type probabilities

A fraction 3/5, a decimal 0.6 or .6, or a percent 60%: all three are the same exact 3/5. The arithmetic uses whole-number numerators and denominators, so 0.1 + 0.2 is exactly 3/10 and a path through four stages of 1/3 is exactly 1/81, not 0.012345679… with rounding error. Decimals at the path ends are rounded to 4 places and marked ≈ when not exact. In a bag, write each count and name, separated by commas: 3 red, 2 blue, 1 green. Branch labels use the first letter when the letters are all different (R, B, G), otherwise the whole word.

What it cannot do

• Maximum of 4 stages and 6 branches per stage (up to 1,296 paths). Bigger trees cannot be read on a page; count with combinations instead.
• Bayes' reversal is not drawn. It will give you P(rain and late), but not the flipped tree for P(rain, given late); divide the path by the sum of the late paths yourself.
• It is not a simulator. Nothing is drawn at random; every number is the exact theoretical probability.
• Branches with probability 0 in a bag (a colour that has run out) are left off rather than drawn as 0.
• Nothing is uploaded. The tree is kept on this device, and the share link carries it in the part after the # sign, which browsers do not send to the server.

Questions

Do you multiply or add on a probability tree?

Multiply along a path, add across paths. A path is one sequence of outcomes, so its probability is the product of its branches. An event that several paths satisfy is the sum of those paths. "At least one red" in the example above is RR + RB + BR = 9/10, which is also 1 − P(BB) = 1 − 1/10.

When are events independent on a tree?

When the second-stage branches are the same whichever way the first stage went. Coin flips and picks with replacement are independent, so "same branches on every stage" fits them. Picks without replacement, or rain making lateness likelier, are dependent: the branches after each first outcome differ. Use the bag mode, or open Edit one point of the tree and give that point its own numbers.