Math Homework. Do It Faster, Learn It Better.

Experimental Probability

Flip a coin 10 times and note down the result. Though you have an equal chance of getting either side, do you always get 5 heads and 5 tails out of 10 trials?

Or, roll a die 60 times and make a note of how many times you get a six. Will you always roll a six 10 times?

In both cases, the answer is No!

Though the theoretical probability of getting heads in the first example is 1 2 , and the theoretical probability of rolling a six in the second example is 1 6 , you may not get heads exactly 1 2 the time, and you may not roll a six exactly 1 6 of the time.

Suppose that, out of 60 rolls of the die, you roll a six 7 times. The fraction 7 60 is called the experimental probability . That is, the experimental probability of an event is the ratio of the number of favorable outcomes to the total number of trials.

With a fair coin or a fair die, you know the theoretical probability ahead of time. Experimental probability is useful in situations where you don't or can't know the probability of an outcome.

Example 1:

Suppose a volleyball team has won 3 of its first 5 matches. Then the experimental probability of its winning the next match is 3 5 = 0.6 .

Suppose there are 11 more matches left in the season. You can use the experimental probability to predict how many of these matches it will win.

0.6 × 11 = 6.6

So, you can expect that the volleyball team will probably win 6 or 7 of its next 11 matches.

Example 2:

The table shows the results, after 20 trials, of drawing one marble from a bag. Each marble drawn is replaced after noting down the color.

Red Blue Green Yellow Black
2 5 6 4 3

What is the expected number of times that a blue marble will be drawn in 500 draws?

We don't know the theoretical probability, because we aren't told how many marbles of various colors there are in the bag. But we do know that the experimental probability of drawing a blue marble is 5 20 or 5 20 . So, the expected number of times that a blue marble will be drawn in 500 trials would be 1 4 × 500 = 125 .