2.2 Events With Equally Likely Results

Last updated: March 8th, 20202020-03-08Project preview

Events with Equally Likely Results

We have said previously that we could calculate the probabilities of events, so now we are going to start by introducing the probability of events in experiments with equally likely results.

Let's think about the example of picking a card from a standard deck of $52$ cards, and let's check what you remember.

First: What would the sample space look like?

$\Omega = \{A\clubsuit, A\diamondsuit, A\spadesuit, A\heartsuit, 2\clubsuit, 2\diamondsuit, ... , K\spadesuit, K\heartsuit \}$

Then, what would be the probability of picking each individual card?

Remember, how many elements are there in $\Omega$?

Since $|\Omega|=52$, for any $\omega\in\Omega$, $P(\omega)=\dfrac{1}{52}$.

Now, imagine we are interested in getting a Queen. The corresponding event would be $E=\{Q\clubsuit, Q\diamondsuit, Q\spadesuit, Q\heartsuit\}$.

How can we calculate $P(E)$?

We have $4$ outcomes that we are interested in, each with a probability of $\dfrac{1}{52}$, so we can add them to get that $P(E)=\dfrac{4}{52}$.

Another way of thinking it, would be that we have $4$ succsessful outcomes out of a total of $52$ options.

Notice that $4$ is the number of elements in $E$, that is $|E|=4$. So, we can write $P(E)=\dfrac{4}{52}=\dfrac{|E|}{|\Omega|}$.

And this will be the case for any experiment with equally likely results, so we can use the formula:

If $\Omega$ has equally likely results, and $E\subseteq\Omega$, then $P(E)=\dfrac{|E|}{|\Omega|}$.

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