4. Probability

A __________ is an arrangement of r objects chosen from n distinct objects without repetition and without regard to order.

1. When you calculate the number of permutations of n distinct objects taken r at a time, what are you counting? Give an example.

Is the Researcher Cheating? You become suspicious when a genetics researcher “randomly” selects numerous groups of 20 newborn babies and seems to consistently get 10 girls and 10 boys. The researcher claims that it is common to get 10 girls and 10 boys in such cases.

a. If 20 newborn babies are randomly selected, how many different gender sequences are possible?

Design Your Own Lottery You have been given the task of creating a new lottery. For each \$1 ticket, the player will select 6 different numbers from 1 to 25 (without replacement), and the only prize will be the jackpot won by players who select the six numbers (in any order) that are later drawn.

a. What is the probability of winning with one ticket?

"Determine the value of each of the following:

b. 0!"

In Exercises 49-53, use counting principles to find the probability.

50. A security code consists of three letters and one digit. The first letter cannot be A, B, or C. What is the probability of guessing the security code on the first try?

In Exercises 49-53, use counting principles to find the probability.

52. A class of 40 students takes a statistics exam. The results are shown in the table at the left. Three students are selected at random. What is the probability that

d. all three students received a B or a C?

In Exercises 49-53, use counting principles to find the probability.

52. A class of 40 students takes a statistics exam. The results are shown in the table at the left. Three students are selected at random. What is the probability that

b. all three students received a C or better?

28. Necklaces You are putting nine blue glass beads, three red glass beads, and seven green glass beads on a necklace. In how many distinguishable ways can the colored beads be put on the necklace?

Mega Millions As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers from 1 to 70 and, in a separate drawing, you must also select the correct single number from 1 to 25.

a. Find the probability of winning the jackpot.

Corporate Officers and Committees The Self Driving Unicycle Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO), and chief human resources officer (CHR). It must also appoint a strategic planning committee with five different members. There are 15 qualified candidates, and officers can also serve on the committee.

b. How many different ways can a committee of five be appointed?

ATM You want to obtain cash by using an ATM, but it’s dark and you can’t see your card when you insert it. The card must be inserted with the front side up and the printing configured so that the beginning of your name enters first.

a. What is the probability of selecting a random position and inserting the card with the result that the card is inserted correctly?

4. The table on the left shows the secondary school student enrollment levels (in thousands by grade) in Oklahoma and Texas schools in a recent year. (Source: U.S. Nation

for Education Statistics)

A student in one of the indicated grades and states is randomly selected. Find the probability of selecting a student who

d. is enrolled in Texas, given that the student is in twelfth grade.

In Exercises 49-53, use counting principles to find the probability.

53. A corporation has six male senior executives and four female senior executives. Four senior executives are chosen at random to attend a technology seminar. What is the

probability of choosing

c. two men and two women?

In Exercises 41-44, perform the indicated calculation.

42. 8P6