1.2. Independent Events and Sample Spaces http://www.ck12.org
your own with your friends. You could say that when you bake a cake or make a cup of hot chocolate, the taste of
these are dependent on what ingredients you use. In the English language, the term dependent means to be unable
to do without, whereas independent means to be free from any outside influence.
What about in mathematics? What do the terms dependent and independent actually mean? This concept will
explore the mathematics of independence and dependence.
Independent Events
In mathematics, the term independent means to have one event not dependent on the other. It is similar to the English
definition. Suppose you are trying to convince your parent/guardian to let you go to the movies on your own. Your
parent/guardian is thinking that if you go, you will not have time to finish your homework. For this reason, you have
to convince him/her that you are independent enough to go to the moviesandfinish your homework. Therefore, you
are trying to convince your parent/guardian that the 2 events, going to the movies and finishing your homework, are
independent events. This is similar to the mathematical definition. Say you were asked to pick a particular card
from a deck of cards and roll a 6 on a die. It does not matter if you choose the card first and roll a 6 second, or vice
versa. The probability of rolling the 6 would remain the same, as would the probability of choosing the card.
Example A
InABCHigh School, 30 percent of the students have a part-time job, and 25 percent of the students from the high
school are on the honor roll. EventArepresents randomly choosing a student holding a part-time job. EventB
represents randomly choosing a student on the honor roll. What is the probability of both events occurring?
EventAis randomly choosing a student holding a part-time job, and eventBis randomly choosing a student on the
honor roll. These 2 events are independent of each other. In other words, whether you hold a part-time job is not
dependent on your being on the honor roll, or vice versa. The outcome of one event is not dependent on the outcome
of the second event. To calculate the probability, you would look at the overlapping part of the Venn diagram. The
region representingAandBis the probability of both events occurring. Let’s look at the probability calculation,
which is done with theMultiplication Rule:
P(A) =30% or 0. 30
P(B) =25% or 0. 25
P(AandB) =P(A)×P(B)
P(AandB) = 0. 30 × 0. 25
P(AandB) = 0. 075In other words, 7.5% of the students ofABChigh school are both on the honor roll and have a part-time job.
Example B
2 coins are tossed one after the other. EventAconsists of the outcomes when tossing heads on the first toss. EventB
consists of the outcomes when tossing heads on the second toss. What is the probability of both events occurring?