# Sample Space

• Construct sample spaces for single-step experiments with equally likely outcomes (ACMSP167)
• Assign probabilities to the outcomes of events and determine probabilities for events (ACMSP168)

## Task: First Down the Mountain

Investigation Sheet A: First Down The Mountain

1. The Game

The following assumes you have run two or three races using the board and dice.

1. Why are there only 11 climbers – from 2 to 12?
2. Why does climber 7 have the furthest to travel in the race?

The major challenge on this worksheet is to decide which climber(s) has the best chance of winning.

2. Computer Software

Use demonstration or automatic demonstration mode.

1. Run 10 races – note the table of results.
2. Run another set of 10 races or trials – has any pattern begun to emerge?

3. Dice Possibilities and Probability

1. Make a list of all 36 possible combinations of the two dice and the totals they give. For exampleย ย ย ย  ย (3,5) = total 8, (1,6) = total 7
2. You will notice that a total of 7 occurs most often, with 6 different ways. From this we can say that the chances of a total of 7 are 6 out of every 36 throws, or 6/36. This is the same as 1/6 and we can write it shorthand as Pr(7) = 1/6.
3. Write down the probabilities or chances for each of the other totals.

4. Run Many Trials

1. Run two or three sets of 100 trials with all 11 climbers.
2. Is 100 trials enough to be confident of naming the ‘best’ climber(s)?
3. Run one set of 1,000 trials and one set of 10,000 trials.
4. Probability: From all your results, write down your best estimate of the long term chances of each climber. Use the % form of presenting these. For example:
 Total 2 3 4 5 6 7 8 9 10 11 12 Per Cent (%) Chance
1. Can you say with confidence which runners have the best and least chances?