Experimental probability

What is wrong with the following statement?

The probability of obtaining a \(6\) when I throw a die is \(\frac{1}{6}\), so if I throw the die \(6\) times I should expect to get exactly one \(6\).

In theory this statement is true, but in practice it might not be the case. Try throwing a die \(6\) times - you won't always get exactly one \(6\).

Question

Kate and Josh each throw a die \(30\) times.

a) How many times would you expect Kate to obtain a \(6\)?

b) How many times would you expect Josh to obtain a \(6\)?

c) What is the total number of sixes you would expect Kate and Josh to obtain between them?

a) In theory, Kate should obtain a \(6\) on \(\frac{1}{6}\) of her throws. Therefore, in theory, you should expect Kate to throw a \(6\) on \(5\) of her \(30\) throws.

b) Josh should also obtain a \(6\) on \(5\) of his \(30\) throws.

c) In total, Kate and Josh have thrown the die \(60\) times. You would expect them to obtain a \(6\) on \(10\) of those throws. It is very unlikely that either Kate or Josh would have obtained exactly five \(6\)s, or that together they would have thrown ten \(6\)s. However, it is more likely that their combined results were closer to the expected outcome (ten \(6\)s) than their individual results.

If an experiment is repeated, the results are not necessarily the same each time. However, as it is more likely that the combined results will be closer to the expected outcome, we can see that if you do a large number of trials you will get a more accurate result.