Branch by branch

The idea

A tree diagram lists all outcomes of a multi-step experiment, a series of independent trials: branches split at each step, each start-to-end path is one complete outcome, and branches can show probabilities.

Draw the possibilities

Toss a coin two times. After the first toss there are 2 possibilities. After the second toss, how many? A drawing can keep track.

A tree starts with one point. How many branches leave it for the first toss?

• 1
• 2
• 4

Grow the tree

Tap the start to split it, and tap each new circle to split it again. Then tap every path to collect the outcomes and find P(HH).

Paths are outcomes

A tree diagram lists all outcomes of a multi-step experiment. Branches split at every step, and each path from the start to the end is one complete outcome.

Two tosses give S = {HH, HT, TH, TT}. Each outcome has probability 1/4. So P(HH) = 1/4 = 0.25 or 25%.

Notes

A tree diagram lists all outcomes of a multi-step experiment. Each path from start to end is one complete outcome, and equally likely outcomes share the probability.

Check yourself

A coin is tossed 3 times. How many outcomes are in the sample space?

Answer: 8

The tree has 2 × 2 × 2 = 8 paths: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

A coin is tossed 2 times. What is P(HH), as a decimal?

Answer: 0.25

P(HH) = 1/4 = 0.25.

A coin is tossed 2 times. What is P(exactly one head), as a decimal? (The event is {HT, TH}.)

Answer: 0.5

The event {HT, TH} has 2 of the 4 outcomes: 2/4 = 0.5.

In a tree diagram, what does one path from the start to the end stand for?

• One toss only. A path covers all the steps of the experiment.
• One complete outcome — correct. Yes. For example the path H then T is the outcome HT.
• The whole sample space. All the paths together make up the sample space.