So far we have computed the probabilities of events that were independent of each other. We saw that getting a certain outcome from rolling a die had no influence on the outcome from flipping a coin, even though we were computing a probability based on doing them at the same time. In this section, we will consider events that are dependent on each other, called conditional probabilities.
conditional probability
The probability the event [latex]B[/latex] occurs, given that event [latex]A[/latex] has happened, is represented as
[latex]P(B | A)[/latex]
This is read as “the probability of [latex]B[/latex] given [latex]A[/latex]”
Conditional Probability Formula
If Events [latex]A[/latex] and [latex]B[/latex] are not independent, then [latex]P(A \text{ and } B) = P(A) · P(B | A)[/latex]
It’s important to remember the conditional probability formula can also be written as [latex]P(A \text{ and } B) = P(B) · P(A|B)[/latex].
Probabilities can be expressed in fraction or decimal form. To convert a fraction to a decimal, use a calculator to divide the numerator by the denominator. Ex. [latex]\dfrac{19}{51}=19 \div 51 \approx 0.3725[/latex]What is the probability that two cards drawn at random from a deck of playing cards will both be aces?
It might seem that you could use the formula for the probability of two independent events and simply multiply [latex]\frac{4}{52}\cdot\frac{4}{52}=\frac{1}{169}[/latex]. This would be incorrect, however, because the two events are not independent.
If the first card drawn is an ace, then the probability that the second card is also an ace would be lower because there would only be three aces left in the deck. Once the first card chosen is an ace, the probability that the second card chosen is also an ace is called the conditional probability of drawing an ace.
In this case, the “condition” is that the first card is an ace. Symbolically, we write this as: [latex]P(\text{ace on second draw } | \text{ an ace on the first draw})[/latex]. The vertical bar “|” is read as “given,” so the above expression is short for “The probability that an ace is drawn on the second draw given that an ace was drawn on the first draw.” What is this probability? After an ace is drawn on the first draw, there are [latex]3[/latex] aces out of [latex]51[/latex] total cards left. This means that the conditional probability of drawing an ace after one ace has already been drawn is [latex]\frac{3}{51}=\frac{1}{17}[/latex]. Thus, the probability of both cards being aces is [latex]\frac{4}{52}\cdot\frac{3}{51}=\frac{12}{2652}=\frac{1}{221}[/latex].