Chapter 7 Chapter 7 Sets & ProbabilitySets & Probability

Section 7.4Section 7.4

Basic Concepts of ProbabilityBasic Concepts of Probability

To determine the probability of the union To determine the probability of the union of two events of two events EE and and FF in a sample space in a sample space SS, we must use the Union Rule for , we must use the Union Rule for Probability, which is based on the Union Probability, which is based on the Union Rule for Sets.Rule for Sets.

Union Rule for ProbabilityUnion Rule for Probability

For any events For any events EE and and FF from a sample from a sample space space SS, ,

P(E P(E F) = P(E) + P(F) – P(E F) = P(E) + P(F) – P(E F) F)

Example: If a single card is randomly drawn from an Example: If a single card is randomly drawn from an ordinary deck of 52 cards, find the probability that it ordinary deck of 52 cards, find the probability that it will be a spade or a face card.will be a spade or a face card.

P(E P(E F) = P(E) + P(F) – P(E F) = P(E) + P(F) – P(E F) F)

P(spade P(spade face) = P(spade) + P(face) – P(spade face) = P(spade) + P(face) – P(spade face) face)

P(spade P(spade face) = face) = 13 13 + + 12 12 - - 3 3 = = 22 22 = = 11 . 11 .

52 52 52 52 2652 52 52 52 26

Possible Outcomes When Two Fair Dice Are Thrown

Example: Two fair dice are thrown. Find the Example: Two fair dice are thrown. Find the probability that the sum of the dice is a number probability that the sum of the dice is a number greater than 9 or that the first die is a 5.greater than 9 or that the first die is a 5.

P(E P(E F) = P(E) + P(F) – P(E F) = P(E) + P(F) – P(E F) F)

P(sum>9 P(sum>9 11stst: 5) = P(sum>9) + P(1: 5) = P(sum>9) + P(1stst:5) – P(sum>9 :5) – P(sum>9 1 1stst:5):5)

P(sum>9 P(sum>9 11stst: 5) = : 5) = 6 6 + + 6 6 - - 2 2 = = 10 10 = = 5 . 5 .

36 36 36 36 1836 36 36 36 18

If events If events EE and and FF are mutually exclusive, are mutually exclusive, then then EE FF = = by defintion; by defintion;

therefore, therefore, PP((EE FF) = 0. ) = 0.

Union Rule for Mutually Exclusive EventsUnion Rule for Mutually Exclusive Events

For mutually exclusive events For mutually exclusive events EE and and F, F,

P(E P(E F) = P(E) + P(F) F) = P(E) + P(F)

An event An event EE and its complement and its complement EE are are mutually exclusive.mutually exclusive.

EE EE = =

The union of an event The union of an event EE and its complement and its complement EE are equal to the sample space. are equal to the sample space.

EE EE = = SS

Complement RuleComplement Rule

PP((EE) + ) + PP((EE ) = 1) = 1

Example: Find the probability that a card drawn from a Example: Find the probability that a card drawn from a standard deck will be larger than a 3. (Aces are high.)standard deck will be larger than a 3. (Aces are high.)

Let Let EE = card is larger than a 3 = card is larger than a 3

Let Let EE = card is a 3 or less (card is a 2 or 3) = card is a 3 or less (card is a 2 or 3)

Find Find PP((EE ). ).

PP((EE ) = ) = 4 + 4 4 + 4 = = 8 8 = = 2 . 2 .

52 52 1352 52 13

Using the Complement Rule, Using the Complement Rule, PP((EE) + ) + PP((EE ) = 1) = 1

PP((EE) + ) + 2 2 = 1 = 1

1313

THUS, THUS, PP((EE) = ) = 11 . 11 .

1313

OddsOddsSometimes probability statements are Sometimes probability statements are given in terms of given in terms of oddsodds, a comparison of , a comparison of PP((EE) + ) + PP((EE ) .) .

ODDSODDSThe The odds in favor odds in favor of an event of an event E E are are defined as the ratio of defined as the ratio of PP((EE) to ) to PP((EE ), ), where where PP((EE ) ) 0. 0.

P(E)P(E) PP((EE ) )

Example: Suppose the sports analysts Example: Suppose the sports analysts say the probability that Alabama will win say the probability that Alabama will win the NCAA National Championship this the NCAA National Championship this year is 3/5. Find the odds in favor of UA year is 3/5. Find the odds in favor of UA becoming National Champs this year.becoming National Champs this year.

P(E) P(E) = = 3/5 3/5 = = 3 3 , , PP((EE ) 2/5 2) 2/5 2

which is written 3 to 2 or 3:2 .which is written 3 to 2 or 3:2 .

Note: Note: ALWAYSALWAYS write odds using “to” or : write odds using “to” or :

Converting Odds to Probability Converting Odds to Probability

If the odds favoring event If the odds favoring event EE are are m m to to nn, , thenthen

P(E) = P(E) = m m and and PP((EE ) = ) = n . n .

m + n m + nm + n m + n

Example: If the odds in favor of Smarty Example: If the odds in favor of Smarty Jones winning the Triple Crown next year are Jones winning the Triple Crown next year are 4 to 5, what is the probability that he will win 4 to 5, what is the probability that he will win the Triple Crown?the Triple Crown?

Triple Crown: Kentucky Derby, the Preakness, and BelmontTriple Crown: Kentucky Derby, the Preakness, and Belmont

The odds indicate chances 4 out of 9 ( 4 + 5) that he The odds indicate chances 4 out of 9 ( 4 + 5) that he will win, sowill win, so

PP(winning) = (winning) = 4 . 4 . 99

Note: Racetrack odds are generally Note: Racetrack odds are generally againstagainst a a horse winning.horse winning.

Empirical vs. Theoretical ProbabilityEmpirical vs. Theoretical Probability

In many real-life problems, it is not possible to In many real-life problems, it is not possible to establish exact probabilities for events. Instead, establish exact probabilities for events. Instead, useful approximations are found by using useful approximations are found by using experimentation or past experiences. This kind experimentation or past experiences. This kind of probability is known as of probability is known as empirical probability.empirical probability.

Theoretical probability is the “true” probability of Theoretical probability is the “true” probability of an event, while empirical probability is the an event, while empirical probability is the probability of the event that had been probability of the event that had been determined through trials or observances.determined through trials or observances.

Example: Charlie Brown knows that the Example: Charlie Brown knows that the probability of getting “tails” when he tosses probability of getting “tails” when he tosses an unbiased coin is 0.5, but when he an unbiased coin is 0.5, but when he actually tosses the coin 10 times, he finds actually tosses the coin 10 times, he finds his probability for getting “tails” is actually his probability for getting “tails” is actually 0.3. 0.3.

Which probability represents the empirical Which probability represents the empirical probability of the event “tails”?probability of the event “tails”?

0.30.3

Properties of ProbabilityProperties of Probability

In a sample space, the probability of each In a sample space, the probability of each individual outcome is a number between 0 individual outcome is a number between 0 and 1.and 1.

0 0 P( P(EE) ) 1 1

The sum of all the probabilities in a sample The sum of all the probabilities in a sample space is always equal to 1.space is always equal to 1.