Thinking Mathematically

Fundamentals of Probability

Computing Theoretical Probability

Sspacesampleinoutcomesofnumber

EeventinoutcomesofnumberEP =)(

Example of Theoretical Probability

What is the probability of getting at most 2 heads when a coin is tossed 3 times?

Solution

The probability of getting at most 2 heads when a coin is tossed 3 times is 7/8

8

7

)(

)()(

8)(

},,,,,,,{

7)(

},,,,,,{

==

==

==

SnEn

EP

SnHHHTHHHTHHHTTTHHTTTHTTTTS

EnTHHHTHHHTTTHHTTTHTTTTE

Example Computing Theoretical Probability

A die is rolled once. Find the Probability of getting a number less than 5.

Solution

The event of getting a number less than 5 can occur in 4 ways: 1, 2, 3, 4.

P(less than 5) =(number of ways a number less than 5 can occur)

(total number of possible outcomes)

= 4/6 = 2/3

Example Probability and a Deck of 52 Cards

You are dealt one card from a standard 52-card deck. Find the probability of being dealt a King.

Solution

Because there are 52 cards, the total number of possible ways of being dealt a single card is 52. We use 52, the total number of possible outcomes, as the number in the denominator. Because there are 4 kings in the deck, the event of being dealt a king can occur 4 ways.

P(king) = 4/52 = 1/13

Example Probabilities in GeneticsEach person carries two genes that are related to the absence or presence of the disease cystic fibrosis. 1 in 25 Americans carries one normal gene and one defective gene. If we use c to represent a defective gene and C a normal gene, such a carrier can be designated as Cc. Thus, CC is a person who neither carries nor has cystic fibrosis, Cc is a carrier who is not actually sick, and cc is a person sick with the disease. One copy of each gene is passed on to the child from the parents. If each parent carries one cystic fibrosis gene, what is the probability that their child will have cystic fibrosis?

SolutionThe table shows the four equally likely outcomes.

Only the cc child will develop cystic fibrosis.

Thus, P(cystic fibrosis) = 1/4

If each parent carries one cystic fibrosis gene, the probability that their child will have cystic fibrosis is 1/4. Second Parent

C cFirst C CC CcParent c Cc cc

Empirical Probability

soccurrenceobservedofnumbertotal

occursEtimesofnumberobservedEP =)(

0.69

1.26

0.69

0.36 Muslim

Catholic

Protestant

GreekOrthodox

Example Computing Empirical Probability

There are approximately 3 million Arab Americans in America. The circle graph shows that the majority of Arab Americans are Christians. If an Arab American is selected at random, find the empirical probability of selecting a Catholic.

Solution

The probability of selecting a Catholic is the observed number of Arab Americans who are Catholic, 1.26 (million), divided by the total number of Arab Americans, 3 (million).

P(selecting a Catholic from the Arab American Population) = 1.26/3 = 0.42

Thinking Mathematically

Fundamentals of Probability