## CONSTRUCTING A PROBABILITY DISTRIBUTION BY SIMULATION (continued)

```       0      0                 200 EXPERIMENTS
0      1
0      2
0      3
0      4
3      5 000
8      6 00000000
9      7 000000000
20      8 00000000000000000000
43      9 0000000000000000000000000000000000000000000
41     10 00000000000000000000000000000000000000000
33     11 000000000000000000000000000000000
17     12 00000000000000000
12     13 000000000000
8     14 00000000
4     15 0000
2     16 00
0     17
0     18
0     19
0     20
```

It appears that we're done enough simulations -- we are starting to see a pretty good pattern in the stemplot. From this graph, we can compute approximate probabilities for different outcomes. To make the below calculations easy, I have listed the frequencies of each outcome to the left of the stemplot.

1. What is an average number of heads in 20 tosses? We see from the graph that the distribution of probabilities is roughly symmetric about 9 or 10 -- either number could be considered an average value.

2. What is the probability of tossing exactly 10 heads in 20 tosses? From the graph, we see that we observed 41 "10 heads" in 200 tosses, so the probability

Probabilty(10 heads) is approx 41/200 = .205

It is interesting to note that, although 10 heads is an average value, it is not a likely value.

3. What are unlikely values? We see from the display that 4 or fewer or 17 or more heads were never observed in our simulation. So the probability of these outcomes must be small.

Page Author: Jim Albert (© 1996)
albert@bayes.bgsu.edu
Document: http://www-math.bgsu.edu/~albert/m115/probability/prob_simulate_3.html