CK-12-Pre-Calculus Concepts

12.7. Basic Probability http://www.ck12.org

P(EC) = 1 −P(E)
You will often be looking at probabilities of two or more independent experiments. Experiments are independent
when the outcome of one experiment has no effect on the outcome of the other experiment. If there are two
experiments, one with outcomeAand the other with outcomeB, then the probability ofAandBis:
P(A and B) =P(A)·P(B)
The probability ofAorBis:
P(A or B) =P(A)+P(B)−P(A and B)
Example A
If you are dealt one card from a 52 card deck, what is the probability that you are dealt a heart? What is the
probability that you are dealt a 3? What is the probability that you are dealt the three of hearts?
Solution:There are 13 hearts in a deck of 52 cards.P(heart) =^1352 =^14
There are 4 threes in the deck of 52.P(three) = 524 = 131
There is only one three of hearts.P(three and heart) = 521
Example B
Dean and his friend Randy like to play a special poker game with their friends. Dean goes home a winner 60% of
the time and Randy goes home a winner 75% of the time.

a. What is the probability that they both win in the same night?

b. What is the probability that Randy wins and Dean loses?

c. What is the probability that they both lose?

Solution:First represent the information with probability symbols.
LetDbe the event that Dean wins. LetRbe the event that Randy wins. The complement of each probability is when
Dean or Randy loses instead.
P(D) = 0. 60 , P(DC) = 0. 40
P(R) = 0. 75 , P(RC) = 0. 25

a.P(D and R) =P(D)·P(R) = 0. 60 · 0. 75 = 0. 45

b.P(R and DC) =P(R)·P(DC) = 0. 75 · 0. 40 = 0. 30

c.P(DCand RC) =P(DC)·P(RC) = 0. 40 · 0. 25 = 0. 10

Example C
If a plane crashes on average once every hundred days, what is the probability that the plane will crash in the next
100 days?
Solution:The naïve and incorrect approach would be to interpret the question as “what is the sum of the probabilities
for each of the days?” Since there are 100 days and each day has a probability of 0.01 for a plane crash, then by this
logic, there is a 100% chance that a plane crashes. This isn’t true because if on average the plane crashes once every
hundred days, some stretches of 100 days there will be more crashes and some stretches there will be no crashes.
The 100% solution does not hold.
In order to solve this question, you need to rephrase the question and ask a slightly different one that will help as an
intermediate step. What is the probability that a plane does not crash in the next 100 days?
In order for this to happen it must not crash on day 1 and not crash on day 2 and not crash on day 3 etc.
The probability of the plane not crashing on any day isP(no crash) = 1 −P(crash) = 1 − 0. 01 = 0 .99.
The product of each of these probabilities for the 100 days is: