Chapter 7 Sets & Probability Section 7.4 Basic Concepts of Probability

To determine the probability of the union of two events E and F in a sample space S, we must use the Union Rule for Probability, which is based on the Union Rule for Sets. Union Rule for Probability For any events E and F from a sample space S, P(E  F) = P(E) + P(F) – P(E  F)

Example: If a single card is randomly drawn from an ordinary deck of 52 cards, find the probability that it will be a spade or a face card. P(E  F) = P(E) + P(F) – P(E  F) P(spade  face) = P(spade) + P(face) – P(spade  face) P(spade  face) = = 22 =

Possible Outcomes When Two Fair Dice Are Thrown

Example: Two fair dice are thrown. Find the probability that the sum of the dice is a number greater than 9 or that the first die is a 5. P(E  F) = P(E) + P(F) – P(E  F) P(sum>9  1 st : 5) = P(sum>9) + P(1 st :5) – P(sum>9  1 st :5) P(sum>9  1 st : 5) = = 10 = 5. P(sum>9  1 st : 5) = = 10 =

If events E and F are mutually exclusive, then E  F =  by defintion; therefore, P(E  F) = 0. Union Rule for Mutually Exclusive Events For mutually exclusive events E and F, P(E  F) = P(E) + P(F) P(E  F) = P(E) + P(F)

An event E and its complement E are mutually exclusive. E  E =  The union of an event E and its complement E are equal to the sample space. E  E = S E  E = S Complement Rule P(E) + P(E ) = 1

Example: Find the probability that a card drawn from a standard deck will be larger than a 3. (Aces are high.) Let E = card is larger than a 3 Let E = card is a 3 or less (card is a 2 or 3) Find P(E ). Find P(E ). P(E ) = = 8 = 2. P(E ) = = 8 = Using the Complement Rule, P(E) + P(E ) = 1 P(E) + 2 = 1 P(E) + 2 = THUS, P(E) = 11. THUS, P(E) =

Odds Sometimes probability statements are given in terms of odds, a comparison of P(E) + P(E ). ODDS The odds in favor of an event E are defined as the ratio of P(E) to P(E ), where P(E )  0. P(E) P(E) P(E ) P(E ) P(E ) P(E )

Example: Suppose the sports analysts say the probability that Alabama will win the NCAA National Championship this year is 3/5. Find the odds in favor of UA becoming National Champs this year. P(E) = 3/5 = 3, P(E) = 3/5 = 3, P(E ) 2/5 2 P(E ) 2/5 2 which is written 3 to 2 or 3:2. which is written 3 to 2 or 3:2. Note: ALWAYS write odds using “to” or :

Converting Odds to Probability If the odds favoring event E are m to n, then P(E) = m and P(E ) = n. P(E) = m and P(E ) = n. m + n m + n m + n m + n

Example: If the odds in favor of Smarty Jones winning the Triple Crown next year are 4 to 5, what is the probability that he will win the Triple Crown? Triple Crown: Kentucky Derby, the Preakness, and Belmont Triple Crown: Kentucky Derby, the Preakness, and Belmont The odds indicate chances 4 out of 9 ( 4 + 5) that he will win, so The odds indicate chances 4 out of 9 ( 4 + 5) that he will win, so P(winning) = 4. P(winning) = 4. 9 Note: Racetrack odds are generally against a horse winning.

Empirical vs. Theoretical Probability In many real-life problems, it is not possible to establish exact probabilities for events. Instead, useful approximations are found by using experimentation or past experiences. This kind of probability is known as empirical probability. Theoretical probability is the “true” probability of an event, while empirical probability is the probability of the event that had been determined through trials or observances.

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

Properties of Probability In a sample space, the probability of each individual outcome is a number between 0 and 1. 0  P(E)  1 0  P(E)  1 The sum of all the probabilities in a sample space is always equal to 1.