Connected ideas

Probability trees and indices

Grow a fair coin tree. Count outcomes and multiply along a path.

Predict, test and explain

Step 1 of 3

Predict

For four fair independent tosses, what is the probability of the specified path HTHT?

\text{specified path: }HHH

Scroll the diagram sideways to see every label.

Ordered outcomes

2^{3}=8

Each toss is fair and independent. Every specified H/T sequence has the same probability. An event such as exactly two heads may combine several paths.

Check the working

Quantity	Value
Assumptions	Independent fair coin tosses
Number of ordered outcomes	8
Selected sequence	HHH
Multiply along this path	$\frac12\times\frac12\times\frac12=\frac1{8}$
Sum of all path probabilities	$8\times\frac1{8}=1$
Exactly two heads	$P=\frac{3}{8}$

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Four fair independent coins are tossed. What is the probability of the specified sequence HTHT?

$P(HTHT)=?$

Hint

Multiply a half for each of the four branches.

Reveal answer

There are sixteen equally likely ordered paths. HTHT is one path.

$P(HTHT)=(\frac12)^4=\frac1{16}$

Build an explanation

Hint 1

A fair independent toss splits every existing path into two equally likely paths.

Hint 2

With n tosses there are 2ⁿ ordered outcomes.

Hint 3

Multiply a half along each branch to find the probability of one specified sequence: (½)ⁿ.

Worked example

This example uses fixed values, separate from the diagram controls.

Three fair independent coins are tossed. Find the probability of HHT, then of exactly two heads.

There are 2³ = 8 equally likely ordered outcomes.
The one specified path HHT has probability ½ × ½ × ½ = ⅛.
Exactly two heads includes HHT, HTH and THH.
Add those three path probabilities: ⅛ + ⅛ + ⅛ = ⅜.

Answer: P(HHT) = ⅛; P(exactly two heads) = ⅜.

Connect this idea

Watch out: One specified sequence is not the same event as a specified number of heads. Equal leaf probabilities here rely on fair, independent tosses.