Chapter 1. Probability

Statistical Applets

Set the probability of heads (between 0 and 1.0) and the number of tosses, then click "Toss". The outcomes of each toss will be reflected on the graph. Check the box to show a line with the true probability on the graph. Click "Reset" at any time to reset the graph.

When you toss a coin, there are only two possible outcomes, heads or tails. On any one toss, you will observe one outcome or another—heads or tails. Over a large number of tosses, though, the percentage of heads and tails will come to approximate the true probability of each outcome.

In this applet, you can set the true probability of heads for your virtual coin, then toss it any number of times. Notice how the proportion of tosses that produce heads can be quite variable at first, but will eventually settle down to the true probability.

eval rand(.2,.8,1)
eval round(1-0.3, 1)
eval round(0.3*10,0)
eval round(0.3*50,0)
eval round(0.3*500,0)

1.

If you set the true probability of heads to 0.3:

  • What should be the true probability of tails?
  • How many heads would you expect to observe after 10 tosses?
2
The probability of tails will be 1 - the probability of heads. To calculate the predicted number of heads, multiply the probability of getting heads on any one toss by the total number of tosses. Try again.
Incorrect. See above for the correct answer.
Great job.

2.

Perform this "experiment" yourself: set the probability of heads to 0.3, then observe how many heads you've gotten after 10 tosses. Enter this number here:

Great.

3.

If the true probability of heads is 0.3, how many heads would you expect to observe after 50 tosses?

After 500 tosses?

2
Correct.
To calculate the predicted number of heads, multiply the probability of getting heads on any one toss by the total number of tosses. Try again.
Incorrect.

4.

Toss the coin some more, and observe how many heads you've gotten after 50 and 500 tosses. At which point (after 10, 50, or 500 tosses) did the actual number of heads agree most closely with the predicted number of heads? Does this result surprise you? Why or why not?
In theory, the observed number of heads should match the predicted number of heads more closely the more tosses you perform. But random events are, well, random: you never know for sure what's going to happen. So don't be surprised to see long "runs" of heads or tails in your experiment that defy the results predicted by the true probabilities.