More tosses, steadier answer

The idea

Experimental probability can differ from theoretical probability, especially when there are few trials, but it tends to get closer to the theoretical probability as the number of trials increases.

Few tosses, many tosses

The theoretical probability of heads is 1/2. But toss a coin 4 times and you may get 3 heads: that is 0.75, far from 0.5. Does the experimental probability ever settle?

If we toss a coin 1000 times, the fraction of heads will be…

• Close to 0.5
• Far from 0.5
• Exactly 0.5

Watch it settle

Press the buttons to toss more and more times. The line shows the running fraction of heads. The dashed line is the theoretical 0.5. Go up to 1000 tosses.

The line settles

Experimental probability can differ from theoretical probability, especially with few trials. As the number of trials increases, it tends to get closer to the theoretical probability. This is the Law of Large Numbers.

Notes

Law of Large Numbers: with more and more trials, experimental probability tends to get closer to theoretical probability. With few trials it can differ a lot.

Check yourself

Which is more likely to be close to 0.5 for a fair coin?

A coin is tossed 20 times and gives 12 heads. What is the experimental probability of heads, as a decimal?

Answer: 0.6

12 ÷ 20 = 0.6. It is a bit more than the theoretical 0.5, which can happen with only 20 tosses.

A student tosses a coin 5 times and gets 4 heads. What does this show?

The Law of Large Numbers says that as the number of trials increases…

• The fraction of heads in 1000 tosses — correct. Yes. With many trials, the experimental probability is usually close to the theoretical value.
• The fraction of heads in 4 tosses. With very few tosses the fraction can easily be far from 0.5.
• Both are equally likely to be close. More trials give a steadier fraction.
• The coin must be unfair. A fair coin can easily give 4 heads in 5 tosses.
• The theoretical probability of heads is 0.8. The theoretical value for a fair coin is still 0.5.
• Very little, since 5 trials are too few — correct. Yes. The experimental probability can be far from 0.5 with so few trials.
• Experimental probability tends to get closer to theoretical probability — correct. Yes.
• Experimental probability gets further from theoretical probability. It is the other way round.
• Theoretical probability changes. The theoretical probability stays the same.