# 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 $\frac{1}{2}$ , and the theoretical probability of rolling a six in the second example is $\frac{1}{6}$ , you may not get heads exactly $\frac{1}{2}$ the time, and you may not roll a six exactly $\frac{1}{6}$ of the time.

Suppose that, out of $60$ rolls of the die, you roll a six $7$ times. The fraction $\frac{7}{60}$ 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 $\frac{3}{5}=0.6$ .

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 $\frac{5}{20}$ or $\frac{5}{20}$ . So, the expected number of times that a blue marble will be drawn in $500$ trials would be $\frac{1}{4}×500=125$ .