BackComputer Variable Names A common computer programming rule was that names of variables must be between one and eight characters long. The first character can be any of the 26 letters, while successive characters can be any of the 26 letters or any of the 10 digits. For example, allowable variable names include A, BBB, and M3477K. How many different variable names are possible? (Ignore the difference between uppercase and lowercase letters.)
Phase I of a Clinical Trial A clinical test on humans of a new drug is normally done in three phases. Phase I is conducted with a relatively small number of healthy volunteers. For example, a phase I test of bexarotene involved only 14 subjects. Assume that we want to treat 14 healthy humans with this new drug and we have 16 suitable volunteers available.
c. If 14 subjects are randomly selected and treated at the same time, what is the probability of selecting the 14 youngest subjects?
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.
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.
c. How does the probability compare to the probability for the old Mega Millions game which involved the selection of five different numbers between 1 and 75 and a separate single number between 1 and 15?
Mendel’s Peas Mendel conducted some of his famous experiments with peas that were either smooth yellow plants or wrinkly green plants. If four peas are randomly selected from a batch consisting of four smooth yellow plants and four wrinkly green plants, find the probability that the four selected peas are of the same type.
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?
Pick 10 Lottery For the New York Pick 10 lottery, the player first selects 10 numbers from 1 to 80. Then there is an official drawing of 20 numbers from 1 to 80. The prize of \$500,000 is won if the 10 numbers selected by the player are all included in the 20 numbers that are drawn. Find the probability of winning that prize.
Matching Probabilities In Exercises 11-16, match the event with its probability.
a. 0.95
b. 0.005
c. 0.25
d. 0
e. 0.375
f. 0.5
16. You toss a coin four times. What is the probability of tossing tails exactly half of the time?
1. When you calculate the number of permutations of n distinct objects taken r at a time, what are you counting? Give an example.
True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
6. 7C5=7C2
In Exercises 7-14, perform the indicated calculation.
9.8C3
In Exercises 7-14, perform the indicated calculation.
12. (10C7)/(14C7)
In Exercises 15-18, determine whether the situation involves permutations, combinations, or neither. Explain your reasoning.
15. The number of ways 16 floats can line up in a row for a parade
In Exercises 15-18, determine whether the situation involves permutations, combinations, or neither. Explain your reasoning.
17. The number of ways 2 captains can be chosen from 28 players on a lacrosse team
20. Skating Eight people compete in a short track speed skating race. Assuming that there are no ties, in how many different orders can the skaters finish?