A possibility diagram lists outcomes of two trials in a rectangular table. Rows describe the first result and columns the second. Count cells as equally likely only when the individual outcomes are equally likely and the trials are independent. Example A fair coin is tossed independently of a fair die. Each cell below records the ordered result: coin first, die second. **Coin / die** | **1** | **2** | **3** | **4** | **5** | **6** H | H,1 | H,2 | H,3 | H,4 | H,5 | H,6 T | T,1 | T,2 | T,3 | T,4 | T,5 | T,6 There are 12 equally likely outcomes. “Head and an even score” includes (H,2),(H,4),(H,6), so its probability is 312= 14. With two distinguishable fair dice, use 36 ordered pairs. The six outcomes summing to 7 are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1), giving probability 636= 16. Different totals are not equally likely: total 2 has only (1,1), while total 7 has six pairs. Do not use the number of distinct totals as the denominator.

A possibility diagram lists outcomes of two trials in a rectangular table. Rows describe the first result and columns the second. Count cells as equally likely only when the individual outcomes are equally likely and the trials are independent.

A fair coin is tossed independently of a fair die. Each cell below records the ordered result: coin first, die second.

There are \(12\) equally likely outcomes. “Head and an even score” includes \((H,2),(H,4),(H,6)\), so its probability is \(\dfrac3{12}=\dfrac14\).

With two distinguishable fair dice, use \(36\) ordered pairs. The six outcomes summing to \(7\) are \((1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\), giving probability \(\dfrac6{36}=\dfrac16\).

Different totals are not equally likely: total \(2\) has only \((1,1)\), while total \(7\) has six pairs. Do not use the number of distinct totals as the denominator.

Possibility diagrams and ordered outcomes

Possibility diagrams and ordered outcomes