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Chapter 16: Probability

This chapter explores the application of probability theory using combinations to solve various card drawing scenarios from a standard deck of 52 cards. It covers fundamental concepts such as finding the probability of drawing specific suits, determining outcomes with multiple cases, and evaluating the likelihood of drawing red or white marbles from a mixed bag. Practical examples include drawing all clubs, no clubs, and exactly one club, alongside similar exercises involving aces and diamonds. The chapter emphasizes counting techniques and the importance of combinations in probability.

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Chapter 16: Probability

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  1. Chapter 16: Probability L16.4 Probability Problems Solved with Combinations

  2. L16.4 Warm Up Three cards are drawn from a well-shuffled standard deck of 52 cards, one after the other and without replacement. • Find the probability of drawing • all clubs b) no clubs • exactly one club (hint: the club can occur on the 1st, 2nd or 3rd drawing) • Evaluate: • b) • Compare your answers to exercises 1 and 2. What do you notice? • Write and evaluate an expression using combinations to find the probability of getting exactly 2 clubs. They are the same!

  3. If E is an event from a sample space S of equally likely outcomes, the probability of event E is: L16.4: Solving Probability Problems w/ Combinations Recall the definition of probability: 0 P(E)  1. • Counting techniques can be used to determine the number of favorable and total number. • Combinations automatically handle situations where multiple cases can occur (e.g., 3 cards with one ace: ANN, NAN & NNA). Example: Three marbles are picked at random from a bag containing 4 red marbles and 5 white marbles. What is the probability that a) all 3 are red b) 2 red marbles c) 1 marble is red d) no red marbles 3+0=3 2+1=3 1+2=3 0+3=3 Notice that the number selected in the numerator is always 3.

  4. Examples 1) Five cards are randomly chosen from a standard deck of 52 cards. Find the probability that the following are chosen a) all 4 aces b) No aces c) exactly 4 diamonds d) four aces and one jack e) at least one ace 2) Three cards are dealt. Find the probability of getting either one ace or two aces. 1 – P(no aces)

  5. 3) A carton contains 200 batteries, of which 5 are defective. If a random sample of 5 batteries is chosen, what is the probability that at least one is defective? 4) Thirteen cards are dealt from a well shuffled standard card deck. What is the probability of getting: a) all cards from the same suit b) 7 spades, 3 hearts, and 3 clubs c) all of the 12 face cards e) at least one diamond P(at least one defective) = 1 – P(no defective) OR P(at least one defective) = P(1 def) + P(2 def) + P(3 def) + P(4 def) + P(5 def) much more complicated… P(all any 1 suit) = P(all ♠) + P(all ♣) +P(all ♥) + P(all ♦) = 4·P(all ♠) P(at least one diamond) = 1 – P(no diamonds)

  6. Should counting always be used? 6. A die is rolled twice a) Find the probability that 2 sixes are rolled b) Find the probability that the face value is greater than 4 and that the second is 2. These are easier to do using probabilities and multiplication.

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