4.3.3. Linking chances to probability

Consider the experiment of tossing a coin once. What are the outcomes? There are only two outcomes – Head or Tail. Both the outcomes are equally likely. Likelihood of getting a head is one out of two outcomes, i.e., $\frac{1}{2}$. In other words, we say that the probability of getting a head = $\frac{1}{2}$. What is the probability of getting a tail?

When we toss a coin, we can only get two results: a Head or a Tail. Since there are no other possibilities, we say these outcomes are equally likely. The probability is calculated by taking the number of favorable outcomes (which is 1 for Heads) divided by the total number of possible outcomes (which is 2, because there are two outcomes: Heads and Tails). Therefore, the probability of getting a Head is $\frac{1}{2}$. By the same logic, the probability of getting a Tail is also $\frac{1}{2}$.

Imagine you are flipping a coin before starting a game to decide who goes first. The uncertainty of whether the result will be Heads or Tails mirrors the idea of probability. Each side is equally likely to land up, similar to the chances you might face in other situations, like choosing between two equally appealing snacks.

Now take the example of throwing a die marked with 1, 2, 3, 4, 5, 6 on its faces (one number on one face). If you throw it once, what are the outcomes? The outcomes are: 1, 2, 3, 4, 5, 6. Thus, there are six equally likely outcomes. What is the probability of getting the outcome ‘2’? $\frac{1}{6}$ ← Number of outcomes giving 2, $6$ ← Number of equally likely outcomes.

When we roll a six-sided die, there are six different outcomes we can get: 1, 2, 3, 4, 5, or 6. Each of these outcomes has the same chance of occurring, hence they are equally likely. To determine the probability of rolling a specific number, such as a 2, we would take the one favorable outcome (the 2 itself) and divide it by the total number of outcomes (which is 6). So the probability of rolling a 2 is $\frac{1}{6}$. Similarly, if we wanted the probability of rolling a 5, it would also be $\frac{1}{6}$, and we can't roll a 7 since it’s not a possibility on a standard die.

Think of a game where you need to roll a die to move forward. Each face of the die represents a different move you can make, and you can’t predict your outcome. This uncertainty is similar to other real-world situations where you want to get a specific outcome from several equally likely options, like picking a colored marble from a bag where each color is represented fairly.

Each outcome of an experiment or a collection of outcomes make an event. For example in the experiment of tossing a coin, getting a Head is an event and getting a Tail is also an event. In case of throwing a die, getting each of the outcomes 1, 2, 3, 4, 5 or 6 is an event. Is getting an even number an event? Since an even number could be 2, 4 or 6, getting an even number is also an event. What will be the probability of getting an even number? $\frac{3}{6}$ ← Number of outcomes that make the event.

In probability, an 'event' is defined not only by specific outcomes but can also be collections of outcomes. For instance, when we toss a coin, we can say getting a Head is one event and getting a Tail is another event. Similarly, throwing a die yields individual numbers as events, but we can also have events like getting an even number, which includes outcomes 2, 4, and 6. To find the probability of this event (getting an even number), we determine the number of favorable outcomes (which are 3) and divide by the total outcomes (which is 6). Thus, the probability of getting an even number is $\frac{3}{6} = \frac{1}{2}$.

Consider a simple game of chance where you toss a coin and call out 'Heads' as your event. If a Tail lands, it’s considered a different event. This is similar to deciding if you’d like to wear an even or an odd number of socks; wearing two socks (an even event) versus three socks (an odd event) can lead to different situations, much like the outcomes in probability.

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