Tossing a Coin

There are always two sides to a coin- Heads and Tails.

Have you ever made an important decision by flipping a coin?

In American football, the captain of each team chooses a side of a coin and then the referee tosses the coin.

The captain who predicts the toss correctly decides about the goal which his team would defend.

Let us see how this works and what is the probability of each side of a coin.

Lesson Plan 

Every coin has two sides: Head and Tail

We denote Head as H and Tail as Tail.

When a coin is tossed, either head or tail shows up. 

The set of all possible outcomes of a random experiment is known as its sample space. Thus, if your random experiment is tossing a coin, then the sample space is {Head, Tail}, or more succinctly, {H, T}.

If the coin is fair, which means that no outcome is particularly preferred, or every outcome is equally likely, then we know that for a large number of tosses, the number of Heads and the number of Tails should be roughly equal. That is, the number of Heads should be roughly 1/2 of the total number of tosses, and so should be the number of Tails. This numerical quantity of 1/2 can be used as a measure of likelihood, or probability.

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• If a coin is fair (unbiased), that is, no outcome is particularly preferred, then we cannot predict heads or tails. Both the outcomes are equally likely to show up.
• If a coin is unfair (biased), that is, an outcome is preferred, then we can predict the outcome by choosing the side which has a higher probability.
  • If the probability of a head showing up is greater than 1/2, then we can predict the next outcome to be a head.
  • If the probability of a tail showing up is greater than 1/2, then we can predict the next outcome to be a tail. 

Suppose that you toss a coin 100 times. Which of the following results is more likely to occur?

Result-1: You obtain 95 Heads and 5 Tails

Result-2: You obtain 48 Heads and 52 Tails

If the coin is a normal everyday coin, in which neither side is particularly prone to showing up more than the other side, you would expect that in a large number of tosses, Heads and Tails should show up roughly an equal number of times. This means that out of the two results above, Result-2 seems to be the more likely one, as the number of Heads is roughly equal to the number of Tails, which concurs with the fact that neither Head nor Tail is a preferred outcome.

On the other hand, in Result-1, the number of Heads is much larger than the number of Tails. Clearly, such a result is extremely biased towards Heads, which is not very likely given that Heads and Tails are equally preferred outcomes. Note that we are not saying that Result-1 is impossible. We are only saying that it is improbable, or unlikely. In other words, the likelihood of Result-1 is much lower than the likelihood of Result-2.

The study of Probability enables us to quantify likelihoods. It enables us to answer questions like: How likely is Result-2? How unlikely is Result-1? And so on.

However, tossing a coin is a random experiment, as you do know the set of outcomes, but you do not know the exact outcome for a particular execution of the random experiment.

 

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Example 1

 

 

A coin is tossed a certain number of times. The relative occurrence of Heads is 0.75. Can we say that the coin is biased towards Heads?

Solution

No, we cannot, because the experiment (tossing the coin) may have been repeated a very small number of times, and thus the relative occurrence in such a scenario will not give the true probability.

No

Example 2

 

 

Coin-A is tossed 200 times, and the relative occurrence of Tails is 0.47. Coin-B is tossed an unknown number of times, but it is known that the relative occurrence of Heads is 0.50. Which coin is fairer?

Solution

It is not possible to comment on the fairness of Coin-B, because the number of times it was tossed is not known. On the other hand, Coin-A seems to be fair, as the relative occurrence of tails over a large number of tosses is almost 1/2.

Coin-A

Example 3

 

 

On tossing a coin twice, what is the probability of getting only one tail?

Solution:

On tossing a coin twice, the possible outcomes are {HH, TT, HT, TH}

Therefore, the total number of outcomes is 4

Getting only one tail includes {HT, TH}

Therefore, the number of favorable outcomes is 2

Hence, the probability of getting exactly one tail is 2/4 = 1/2

1/2

Here are a few activities for you to practice.