762 CHAPTER 6 PROBABILITY MODELS EXERCISES ACTIVITY 6.7 1. Compute each of the following: 100! a. 5! I). 98! c. 9P 9 ~~ d. np 9 g- 8Q e. 10^4 6^4 " 285^1 f-, 2 c 5 ' sq ' sq 2. How many different ways are there to arrange twelve books on a shelf? 3. How many ways are there to select a committee of five from a club with 30 members? 4. How many different four-digit numbers can be made using the digits 1, 2, 3, 4, 5, 6 if no digit can be used more than once? 5. How many ways are possible to select a president and vice president from an association with 5400 members? 6. How many different words can be made by rearranging the letters of the word SYMMETRY? 7. If a coin is flipped 50 times, in how many ways could there be exactly two tails? 8. A lottery ticket has 54 numbers, from which the player chooses six. a. What is the probability that all six numbers are correct, and the player wins the big jackpot? b. What is the probability that the player gets five of the six numbers correct?
ACTIVITY 6.7 SELECTING AND REARRANGING THINGS 763 9. A club with 46 members, of which 20 are girls, needs to form a committee of six, to be composed of the same number of boys as girls. In how many ways can this committee be selected? {Hint: Calculate two different numbers of combinations, then apply the multiplication principle.) 10. The number of combinations for any size collection makes an interesting pattern, usually called Pascal's triangle, after the French mathematician/philosopher Blaise Pascal (1623-1662). This pattern was introduced back in the exercises for Activity 1.2. Each row of the triangle has one more number than the previous row, each number the result of adding the two numbers immediately above it. The first and last numbers are always one. a. Identify the pattern and then complete the next three rows of Pascal's triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 b. Each row actually contains the number of combinations of n objects taken r at a time, where n is the number of the row (starting with 0) and r is the position in the row (again, starting with 0). For example, 1, 3, 3, 1 are 3C 0, 3 C(, 3C 2, 3C3. Which number in Pascal's triangle corresponds to 7C 4? Verify with your calculator. c. When displayed in this form, the symmetry of the combinations is plain to see. If you know the value of 30C g, what else is it equal to?
764 CHAPTER 6 PROBABILITY MODELS ACTIVITY 6.8 How Many Boys (or Girls)? OBJECTIVES 1. Recognize components of binomial experiment. 2. Calculate binomial probabilities. In a family with five children, how unusual would it be for all the children to be boys? Or by the same token, all girls? Such questions are relevant for many fami- "' lies in cultures around the world, especially when one gender is favored over another. Suppose a family does have all boys. What is the probability of such an event? 1. Assume the birth of a boy or a girl is equally likely. What is the probability that I a single child is born a boy? 2. Now suppose a family has one boy already. What is the probability that their next child will also be a boy? The probabilities you just determined are for a single child. You are really interested in the probability that for a family with two children, both are boys. 3. For a family with two children, what are the possible outcomes? Provide a tree diagram or simply list the possibilities. 4. a. Use the sample space in Problem 3 to determine the theoretical probability of the event that a family with two children has two boys. b. Assume that the individual outcomes of the two births are independent (i.e., the outcome of the second birth in no way depends upon the outcome of the first birth). Use the multiplication principle of probability to determine \ the probability that a two-child family has two boys. 5. What is the theoretical probability of the event that a family with two children has two girls? 6. What is the theoretical probability of the event that a family with two children has one boy and one girl? 7. At this point, you have determined the probabilities of all the possible outcomes for a two-child family. Letting the random (independent) variable x represent the number of boys in a family with two children, complete the following table for and verify the properties of the probability distribution. Recall: 0 ^ P(x) 1 for all x in the sample space and the sum of the probabilities of all x in the sample space must equal 1.
ACTIVITY 6.8 HOW MANY BOYS (OR GIRLS)? 765 Note that P{\ boy and 1 girl) =. This is because there are two ways the family could have one boy, either first born or second born. If you wanted the probability of "the first born is a boy and the second born is a girl," the probability is the product of the probabilities of independent events. Binomial Probability Distribution There are many situations where each individual outcome can go one of two ways: boy or girl, head or tail, right or wrong, success or failure. Assuming the two outcomes have well defined and fixed probabilities, combinations of such outcomes define a binomial probability distribution. To determine an individual binomial probability, it is necessary to count how many ways an event can occur (like the two distinct ways a family with two children could have one boy). For the moment, let's consider a different binomial (two outcome) situation, flipping a coin. Assuming fairness (head and tail equally likely, as we had assumed for boy and girl), P(head) = P(tail) = 0.5. 8. Consider flipping a fair coin three times. Let x = the number of heads, record the sample space for this experiment (as a tree, chart or list), and fill in the table for this binomial probability distribution. 9. Repeat the same process for flipping a coin four times.
766 CHAPTER 6 PROBABILITY MODELS There are several shortcuts for determining the number of ways a certain number ; of heads can come up. In flipping a coin four times, it is not too difficult to simply a list the different ways exactly two heads could result: hhtt, htht, huh, thth, thht, anda hhtt. If you can agree that these are all different, and there are no more possible, then I the theoretical probability of getting exactly two heads when four coins aref flipped is = -z. (Is this what you got in Problem 9?) lo o But suppose you flipped a coin 100 times, or even only ten times. To determine the J number of ways to get exactly five heads by listing them, or drawing a tree, would be tedious. There is, however, a formula for calculating these numbers, which are called binomial coefficients. They are the very same combinations you learned I about in Activity 6.7. Example 1 Consider Problem 9, in which you flipped a coin four times. How many J ways can exactly two heads occur? SOLUTION This is sometimes called the "number of combinations of four things taken two at a time." There are two types of notation used, I 1 and 4C 2. The computation is _ 4-3-2-1 _, 4t - 2 (2-1)(2 1) The number of ways to get exactly three heads on five flips would be 5Ü3 5-4-3-2-1 (3-2- 1)(2-1) = 10. In general, n\ " Cr ri'-"(n-r)! n{n - 1) - 3 2-1 (r(r- l)--3-2-l)((n- r){n - r - 1) 3-2-1) This is a rather complicated formula, but binomial coefficients can be easily computed on your calculator. (See Appendix A.) 10. Calculate the number of ways to get exactly four heads when a coin is flipped five times, by applying the formula and by checking on your calculator.
ACTIVITY 6.8 HOW MANY BOYS (OR GIRLS)? 767 11. Now determine the binomial probabilities for the number of heads when a coin is flipped five times. 12. Experimentally, you can see if your probability distribution seems reasonable by actually flipping coins. Instead of flipping an individual coin five times, use five coins, shake them up in your cupped hands, and drop them on the table, recording the number of heads. Repeat this ten times, recording your results in a frequency table. How close are your relative frequencies to the theoretical probabilities you calculated in Problem 11? X TEN REPETITIONS FREQ. REL. FREQ. FREQ. ENTIRE CLASS REL. FREQ. 13. Record the frequencies and relative frequencies for the entire class. Did you observe what you expected, with regard to the theoretical probabilities? Explain. 14. It is also possible to simulate many repetitions of the experiment on a computer or your calculator. (See Appendix A.) Simulate flipping five coins 500 times on your calculator, by generating a list of 500 random integers between 0 and 5, from a binomial distribution. Record your results in the table, and then record the collective total for your entire class. Comment on the results, compared to your previous work. ^^^^H 500 REPETITIONS ENTIRE CLASS x 1 FREQ. REL. FREQ. 1 FREQ. 1 REL. FREQ.
768 CHAPTER 6 PROBABILITY MODELS f/vtu ^' ptñi, ^ ' s a ' s0 P ossm ' e t0 determine any binomial probability directly from a tablei or your calculator. (See Appendix A.) Use your calculator to verify trs theoretical probabilities you calculated in Problem 11. In general, a binomial probability is calculated using the following formula: P(x)=\ y<f-\ where n is the number of independent trials, x is the random variable, what is being counted, p is the probability of a success (the outcome x is counting), q is the probability of a failure (the other outcome) For example, in the previous Problems 11-15, n = 5, p = q = 0.5. The probability j of exactly three heads (successes) out of five coin flips is PO) - (*)M*S» 3 = (3 5 2 4 ;j:( 2? 1) '" 5-03125. I 16. How close did your final relative frequency for three heads come to this! theoretical probability? 17. a. Apply the formula to determine the probability that a two-child family has] both boys. b. Apply the formula to determine the probability that among eight children] exactly five will be boys. Verify your result by finding directly with your J calculator. Because you assumed a fair coin, and the birth of a boy and girl are equally likely,] the probability of a success and the probability of a failure were both 0.5. Usually j these probabilities are different, as you might have guessed from the formula. 18. Suppose that 10% of all electrical switches are defective. a. What is the probability of selecting a defective switch in a single selection b. What is the probability of selecting a good switch in a single selection? c. If you select five switches, what is the probability that exactly one m defective? Apply the formula and then verify directly on your calculator.