Calculate the probability of tossing a coin 20 times and getting the given number of heads

There are two different types of probability that we often talk about: theoretical probability and experimental probability.

Theoretical probability describes how likely an event is to occur. We know that a coin is equally likely to land heads or tails, so the theoretical probability of getting heads is 1/2.

Experimental probability describes how frequently an event actually occurred in an experiment. So if you tossed a coin 20 times and got heads 8 times, the experimental probability of getting heads would be 8/20, which is the same as 2/5, or 0.4, or 40%.

The theoretical probability of an event will always be the same, but the experimental probability is affected by chance, so it can be different for different experiments. The more trials you carry out (for example, the more times you toss the coin), the closer the experimental probability is likely to be to the theoretical probability.

Maybe you could try tossing a coin 20 times to see how close your experimental probability is to the theoretical probability.

PROBABILITY 2 HEADS & TAILS ACTIVITIES 1 AND 2

Fig.1 The U.S. presidential campaign.

Fig.2 The maths might not be quite right but can you see the point the add makes?

Experimental Probability

Probability is measured on a decimal scale of 0 to 1, but students are also encouraged to use fractions and percentages as this links in nicely with Number. 0 represents an event which is impossible and 1, an event which is certain. The probability or chance that there will be sunrise tomorrow is a certainty even if we can't see it due to the nature of the weather. The probability of me flying by running around flapping my arms rapidly would be represented by 0, impossible. The probability of getting heads by tossing a fair coin would be 0.5 (assuming it doesn't land on its edge).

Fig.3 This scale shows how the probability of an event can be described.

Equally likely activity

Working in pairs, write out a sequence of Heads and Tails that you might think would result from tossing a coin 20 times. For example, you might think HTTHTHHTTTH etc.

Fig. 4 Results of one real trial of tossing a coin 20 times

ACTIVITY 1

Write out your own 'random generated' H/T combination. Repeat it 20 times, e.g. HHTTHTHTHHHTTTTHTTHH.

Fig.5 Which data set is fake, left or right?

What many people don't realise is that consecutive tails and heads occur more frequently than expected. This means that when students try to create their own data they usually don't record more than 3 heads or tails in a row.

activity 2

The aim of this activity is to calculate the experimental probability of obtaining heads from a coin toss. You only have to be aware of the concept of the running average at this stage. Because this activity is random, we should get slightly different results between the groups. We know from theory that the probability is 0.5 or 1/2. Toss a coin 50 times and record the results in a frequency table like the one shown below. You can use the simple frequency in Fig. 7 if you like but it doesn't explain the running average concept. Notice each toss of the coin is called a trial, the outcome of the experiment is either heads or tails, we will assume a coin landing on its edge is not valid data and can be ignored so toss the coin again. We will also assume the coin is fair and tossed in an unbiased manner. Consecutively number the heads in column 3 as shown below. To get the running total, make a fraction using column 4 as the numerator and column 1 as the denominator. You can use a calculator to work out the decimal value as shown below in the table on the right in Fig.6 

Fig.6 For graph drawing its easier to use decimals for running averages as shown on the right hand table.

If you find the running average concept confusing then use the simplified frequency table in Fig.7 below.

Fig. 7 Simple version of frequency table.

Calculating the probability. 

Experimental probability =

For example if you got 24 heads after 50 trials,

the experimental probability =

Once you are done, create a simple bar chart like the one below, showing the distribution of Heads and Tails. In my case I got 24 Heads and 26 Tails. That gives a total of 50 trials.

Fig. 8 Bar Chart

activity 3 running average

To calculate the running average use the information from the running total column of the frequency table in Fig.6. I used Excel to create these charts - you are allowed technology to do this.

Fig.9 Long running average from the above data

So how does this work? Referring to the frequency table of Fig.6 you can see I got a T first and no H, so the graph in Fig.9 above starts at zero. My next flip of my coin gave me my first H. Over two throws I had now one T and one H which is 1/2 or 0.5 so the second point is 0.5 on the graph. My next H happened on the 5th flip of the coin, so far that's two H's after five flips or 2/5 = 0.4. Notice the average is 'moving' as we move through the trials. That's why it is called a running average.

What does the running average tell us?

The theoretical probability as discussed is 1/2 or 0.5. However, we can't always arrive at this conclusion early on in an experiment after just a few coin flips. We have also learned that it is quite possible to get 5, 6, 7 or more H's in a row. In my own example, the fact that I had only 2 H's after 5 flips of the coin is indicated by the 'violent' swings of the lines of Fig.9. In the long run however, I would expect the coin flips to 'even or smooth out'. It might take 500 or 1000 flips of the coin to achieve this. Below is a graph showing the long term running average of 500 flips of a coin. Notice the wide swings before it begins to 'settle down".

gamblers Fallacy

Essentially the Gambler's Fallacy is based on a belief that if some random event happens more frequently that usual then it will become less frequent in future. An example might be that a gambler sees five Heads's in a row from a coin flip. When asked what he thinks the next flip will be he will say it must be a Tail. This is also known as Monte Carlo's Fallacy.

One year, at the casino in Monte Carlo, black came up a record twenty-six times in succession on the roulette table. This produced a mad gamblers panic to place their bets on red around about the time the ball landed on the 15th black. It only seemed natural that this would occur. However  the little white ball landed on a black number again and the casino won more money than it usually did. No matter how many times you get consecutive heads in a coin toss, the next coin has a 0.5 probability of  being either heads or tails!

What is the probability of getting heads if you flip a coin 20 times?

What is the probability of getting heads 20 times in a row?

How many possible outcomes are there when a coin is tossed 20 times?

How many times do you think heads will occur if a coin is tossed 20 times?