6.1 – Calculating Probabilities

What is Probability?

Probability is a quantitative, or numerical, expression of the chance of something happening. For example, let’s say that we flipped a coin. We can state that there is a 50% chance of the coin landing on “heads.” While we don’t know for sure that the coin will come up “heads” we can make an educated guess about the probability of that event happening.

The calculation of probability is relatively straightforward. Count the number of outcomes that you are looking for, and then divide by the total number of possible outcomes:

[latex]P(\text{targeted outcome})= \frac{\text{total number of targeted outcomes in set}}{\text{total number of all possible outcomes in set}}[/latex]

So when we flip a coin and we want to know the probability of it coming up “heads” we simply count the number of outcomes that are “heads” (1), and then divide by the total number of outcomes possible (“heads” and “tails” or 2):

[latex]P(heads)= \frac{\text{total number of sides of the coin that are "heads"}}{\text{total number of sides of the coin}}=\frac{1}{2}=0.50[/latex]

While the calculation of probability is pretty simple, the difficulty comes in making sure you count your outcomes correctly. Let’s say we are working with a deck of cards and we want to know the probability of randomly picking an Ace of Spades from the deck. To calculate this probability, we first count the number of cards in the deck that are Ace of Spades, which is one. Then we count the total number of cards, which is 52. Thus our probability is:

[latex]P (\text{Ace of Spades}) = \frac{\text{Number of Ace of Spades in the Deck}}{\text{Total Number of Cards in the Deck}} = \frac{1}{52} = 0.01923[/latex]

There is a 1.923% chance that we draw an Ace of Spades.

What about the probability of drawing any Ace? To calculate this probability we first count the number of Aces in the deck, which is four. Then we count the total number of cards, which is 52. Thus our probability is:

[latex]P (\text{Ace}) = \frac{\text{Number of Aces in the Deck}}{\text{Total Number of Cards in the Deck}} = \frac{4}{52} = 0.0769[/latex]

There is a 7.69% chance of drawing one of the Aces.

What about the probability of drawing any of the Spades? To calculate this probability we first count the number of Spades in the deck, which is 13. Then we count the total number of cards, which is 52. Thus our probability is:

[latex]P (\text{Spades}) = \frac{\text{Number of Spades in the Deck}}{\text{Total Number of Cards in the Deck}} = \frac{13}{52} = 0.25[/latex]

There is a 25% chance of drawing one of the Spades.

As you can see from these card examples, you want to make sure that the numerator (top) of the equation has a count of all the possible outcomes that fit your target. In the last example, we were looking for any of the Spades. That includes the following 13 cards: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A. So now we know that if we pull one card at random from the deck, it has a 25% chance of being one of these Spades.

Conditional Probability

A conditional probability is a probability of an event happening based upon a particular condition, or other event happening. For example, we can ask the question, “What is the probability of drawing an Ace of Spades, if we know that the card is a Spade? We want to know the probability of one event (drawing an Ace of Spades) based upon a particular condition (if the card drawn is a Spade).

The calculation of the probability uses the same formula, but we have to be careful about counting things correctly based on the conditions. In our example, we first want to know how many outcomes fit the targeted outcome. Since there is only one Ace of Spades in the deck, the answer is 1. Next, we want to know how many possible outcomes are there. Since our condition states that we have drawn a Spade, we can then determine that there are 13 possible cards that we could have drawn: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A. Thus, the denominator (bottom) of our equation is 13.

[latex]P (\text{Ace of Spades} | \text{Spades}) = \frac{\text{Number of Ace of Spades in the Deck}}{\text{Total Number of Spades in the Deck}} = \frac{1}{13} = 0.0769[/latex]

There is a 7.69% chance that the card we selected is the Ace of Spades.

Note that the depiction of the probability, “P (Ace of Spades | Spades),” includes in the parentheses the targeted outcome followed by the condition, with a line in between. This is read as “the probability of an Ace of Spades given a Spades.”