Transcript Probability
Probability Counting Outcomes and Theoretical Probability What is Probability? • the relative frequency with which an event occurs or is likely to occur • typically expressed as a ratio Counting Principle • to determine the total number of possibilities that can occur in an event Example #1 • A store sells caps in three colors (red, white, and blue), two sizes (child and adult), and two fabrics (wool and polyester). • How many cap choices are there? • (colors)(sizes)(fabrics) • (3)(2)(2) = 12 choices Example #2 • How many three letter monograms are possible in the English language? • (1st letter) (2nd letter) (3rd letter) • (26)(26)(26) • 17576 Theoretical Probability • number of favorable outcomes number of possible outcomes why is it called theoretical? Example #3 • What is the probability of winning the Play-4 lottery if you purchase two tickets with different numbers? (1st digit)(2nd digit)(3rd digit)(4th digit) (10)(10)(10)(10) 10000 2:10,000 or 1:5,000 Independent Events • Events that do not have an affect on one another – – – – Tossing a coin multiple times Rolling a die multiple times Repeating digits or letters Replacing between events Dependent Events • One event happening affects another event happening – Not replacing between events – No repeating digits or letters P(A, then B) • The probability of event A occurring then the probability of event B occurring • Total probability is determined by multiplying the individual probabilities Example #1 • You choose a card from a regular deck of playing cards. After returning it to the deck, you choose a second card. What is the probability that the first card will be red and the second card will be a seven? • 52 cards in the deck • 26 red and 26 black • 4 sevens 26 52 1 2 4 52 1 26 1 13 Example #2 • You choose a card from a regular deck of playing cards. Without returning it to the deck, you choose a second card. What is the probability that the first card will be a red face card and the second card will be a seven? • • • • 52 cards in the deck 26 red and 26 black 12 face cards 4 sevens 6 52 4 3 26 51 4 51 2 221 Example #3 • There are five girls and two boys seated in a waiting room. What is the probability that the first person called will be a girl and the second one called will be a boy? • 5 girls • 2 boys 5 7 5 7 2 6 5 21 1 3 Example #4 • There are five girls and two boys seated in a waiting room. What is the probability that the first person called will be a boy and the second one called will be a girl? • 5 girls • 2 boys 2 7 5 21 5 6 Permutations • determining the number of arrangements of items when the order is important • nPr – P → permutations – n → number of objects – r → number chosen Example #1 • 8 people wish to buy tickets for a concert. In how many ways could the first five members get in line? • 8P5 = (8)(7)(6)(5)(4) • 8P5 = 6720 Example #2 • How many arrangements of four books can be made from a stack of nine books on a shelf? • 9P4 = (9)(8)(7)(6) • 9P4 = 3024 Combinations • determining the number of arrangements of items when the order is not important • nCr – C → combinations – n → number of objects – r → number chosen Example #3 • You have five choices of sandwich fillings. How many different sandwiches can you make using three of the fillings? • 5C3 = 5P3/3P3 • 5C3 = (5)(4)(3)/(3)(2)(1) • 5C3 = 10