Probability of Multiple Events
Probability of Multiple Events
If events are independent events, the probability of one occurring does not affect the probability of the other event occurring. Rolling a die and getting one number does not change the probability of getting any particular number on the next roll. The number of faces has not changed, so these are independent events.
The probability of two independent events occurring is the product of the two events’ probabilities:
P(A and B) = P(A) × P(B)
A bag contains five marbles: two red, one blue, one green, and one orange. What is the probability of choosing a red marble and rolling an even number on a six-sided die?
Since there are two red marbles and a total of five marbles, there is a \(\frac{2}{5}\)chance of choosing a red marble. There are three even numbers on a die, so the chance of rolling an even number is \(\frac{3}{6} = \frac{1}{2}\).
\(\frac{2}{5}×\frac{1}{2}=\frac{1}{5}\)
The probability of one independent event or another occurring can be found using the following formula:
P(A or B) = P(A) + P(B) – P(A and B)
When selecting a single card from a standard deck of 52 cards, what is the probability of drawing either a diamond or a face card?
P(A or B) = P(A) + P(B) – P(A and B)
Let A be the event of drawing a diamond → \(\frac{13}{52}\)
Let B be the event of drawing a face card → \(\frac{12}{52}\)
There are three cards that are both a diamond and a face card → \(\frac{3}{52}\)
\(P(A\ or\ B) = \frac{13}{52}\ + \frac{12}{52}\ −\ \frac{3}{52} = \frac{11}{26}\)
If events are dependent events, the probability of one event occurring changes the probability of the other event occurring. For example, if cards are drawn from a deck without being replaced, the probability of drawing a specific card changes after each draw.
Conditional probability is the probability of an event occurring given that another event has occurred. The notation P(B|A) represents the probability that event B occurs, given that event A has already occurred (it is read “probability of B, given A”). Conditional probability is used to find the probability of multiple dependent events:
P(A and B) = P(A)P(B│A) = P(B)P(A|B)
A fish tank contains three red fish and one blue fish. What is the probability of catching two red fish in succession without replacing the first one?
probability of catching one red fish = \(\frac{3}{4}\)
probability of catching one red fish if one has already been caught = \(\frac{2}{3}\)
\(P(A\ and\ B) = \frac{3}{4}\ × \frac{2}{3} = \frac{1}{2}\)