A common way for two people to settle a frivolous dispute is to play a game of rock-paper-scissors. In this game, each person simultaneously displays a hand signal to indicate a rock, a piece of paper, or a pair of scissors. Rock beats scissors, scissors beats paper, and paper beats rock. If both players select the same hand signal, the game results in a tie.
Two roommates, roommate A and roommate B, are about to go cruising with a mutual friend and are arguing over who gets to sit in the front seat. Roommate A suggests a game of rock-paper-scissors to settle the dispute.
1. Consider the game of rock-paper-scissors to be an experiment. How many outcomes are in the sample space of this experiment?
Assume that both roommates' choices of rock, paper, or scissors are random, independent, and equally likely. Let RA, RB, PA, PB, SA, and SB denote roommate A choosing rock, roommate B choosing rock, roommate A choosing paper, roommate B choosing paper, roommate A choosing scissors, and roommate B choosing scissors, respectively. Additionally, let the sample point consisting of roommates A and B both selecting rock be denoted parenthetically as (RA, RB), and so on for the other possibilities.
Define event A as the event that roommate A wins the game and gets to sit in the front seat.
2. Event A is composed of the following sample points:
A = {(RA, RB), (RA, SB), (PA, RB), (SA, PB)}
A = {(RA, PB), (PA, SB), (SA, RB)}
A = {(RA, SB), (PA, RB), (SA, PB)}
A = {(PA, RB), (SA, PB)}
3. What is the probability of event A?
Let event C be the event that the game ends in a tie.
4. Event C is composed of the following sample points:
C = {(RA, RB), (PA, PB), (SA, SB)}
C = {(RA, RB), (RA, PB), (RA, SB), (PA, RB), (PA, PB)}
C = {(RA, RB), (RA, PB), (RA, SB), (PA, RB), (PA, PB), (PA, SB), (SA, RB), (SA, SB)}
C = {(RA, RB), (RA, PB), (PA, PB), (SA, PB), (SA, SB)}
5. What is the probability of event C?
SCRABBLE® is a word game in which two to four players score points by forming words from individual lettered tiles on a game board marked with a 15-by-I5 grid. The words are formed across and down in crossword-puzzle fashion and must appear in a standard dictionary. There are 100 tiles in all, including 2 blank tiles that serve as wild cards and can take on the value of any letter.
Frequency of the SCRABBLE Letter Tiles
6. Fill in the probabilities. Enter your answers as decimals, and assume that you are sampling with replacement.
(a) What is the probability of drawing a D?
(b) What is the probability of drawing a K?
(c) What is the probability of drawing an A?
7. In the game of SCRABBLE, you select letters from the group in the pot that are not already on the board or in your hand or someone else's. This would be sampling letters without replacement. Suppose that instead of doing this, you select a letter from the pot, write it down, and then return it to the pot. In other words, suppose you sample letters with replacement. Every time you select a new letter, you write it down next to the letter you drew previously. Because you are sampling the pot with replacement, each draw of a letter is an independent event such that the Multiplication Rule for Independent Events applies.
(a) Select a letter, write it down, and return the letter to the pot. Do this three times.
8. When you reach into the pot to select your letters, your result is based on a simulation. In a simulation, random events are modeled in such a way that the simulated outcomes closely match real-world outcomes. In this simulation, your result is simulated based on the frequency distribution of SCRABBLE letters. (Note: Because the results of this experiment are based on a random simulation, Aplia is not responsible for any inappropriate words that may result.)
Before selections your titles what is the possibility that your selection (in order) spells the word "jet" of the blank tile as a wild card? Give your answer to six decimal places.
9. Based on the preceding probability, you _____________ to spell the word under consideration.
10. Repeat your experiment, but this time select, write, and return the letter four times. Again, your results will be generated based on the frequency distribution. Before you select your tiles, what is the probability that the four letters you write down (in order) will spell "cake" (without the use of a blank tile as a wild card)? Give your answer to eight decimal places.
11. Based on the preceding probability, you ____________ to spell the word under consideration.
In the game of Pick-A-Ball, there are 10 colored balls: 3 red, 4 white, and 3 blue. The balls have been placed into a small bucket, and the bucket has been shaken thoroughly. You will be asked to reach into the bucket without looking and select 2 balls. Because the bucket has been shaken thoroughly, you can assume that each individual ball is selected at random with equal likelihood of being chosen.
It is common in games of chance to be rewarded for obtaining unlikely results. For example, in the game of poker, hands that are less likely to occur tend to trump hands that are more likely to occur. That same concept applies in this game,
It costs $10 to play Pick-A-Ball, and your prize is the reciprocal of the probability of obtaining your exact sequence of results, expressed in dollars. For example, if the probability of obtaining your exact sequence of results is 0.10, your prize is 1/0.10 = $10, and you break even. Therefore, to make a positive profit from playing Pick-A-Ball, the probability of obtaining your results must be lower than 0.10.
When you reach into the bucket, your result is based on a simulation. In a simulation, random events are modeled in such a way that the simulated outcomes closely match real-world outcomes. In this simulation, your result is simulated based on the probabilities of obtaining a ball of a particular color.
Now, close your eyes! Reach into the bucket, and pick a ball. (Click the red Pick-A-Ball! icon to simulate reaching into the bucket and drawing your ball.)
12. What is the probability of selecting the color of ball that you just selected? (Round your answer to four decimal places.)
Don't put your first ball back into the bucket. Now, reach in (again, no peeking!), and pick your second ball.
13. What is the probability of selecting the color of ball that you just selected? (Round your answer to four decimal places.)
14. What is the probability of Obtaining your exact sequence of results? (Round your answer to four decimal places.)
A common way for two people to settle a frivolous dispute is to play a game of rock-paper-scissors. In this game, each person simultaneously displays a hand signal to indicate a rock, a piece of paper, or a pair of scissors. Rock beats scissors, scissors beats paper, and paper beats rock. If both players select the same hand signal, the game results in a tie.
Two roommates, roommate A and roommate B, are expecting company and are arguing over who should have to wash the dishes before the company arrives. Roommate A suggests a game of rock-paper-scissors to settle the dispute.
Consider the game of rock-paper-scissors to be an experiment. In the long run, roommate A chooses rock 36% of the time, and roommate B chooses rock 22% of the time; roommate A selects paper 32% of the time, and roommate B selects paper 25% of the time; roommate A chooses scissors 32% of the time, and roommate B chooses scissors 53% of the time. (These choices are made randomly and independently of each other.)
15. The probabilities were assigned using the ______________________.
Define event B as the event that roommate B wins the game and thus does not have to wash the dishes. What is P(B), the probability of event B?
- P(B) = 0.33
- P(B) = 0.67
- P(B) = 0.43
- P(B) = 0.22
Let event C be the event that the game ends in a tie.
What is P(C), the probability of event C?
- P(C) = 0.33
- P(C) = 0.50
- P(C) = 0.86
- P(C) = 0.55
Define event A as the event that roommate A wins the game and thus does not have to wash the dishes. What is the probability that roommate A wins the game?
- P(A) = 0.43
- P(A) = 0.34
- P(A) = 0.37
- P(A) = 0.39
What is the complement of event B?
- BC = event B
- BC = event C
- BC = event A
- BC = event A or event C
16. The probability of BC is ______.
Surveys conducted for the Pew Internet & American Life Project suggest that communication patterns vary across three groups of teenagers: social networkers, content creators, and multichannel teens (teens who use the Internet, instant messaging, text-messaging cell phones, and social networks).
In one survey, Internet-using teenagers were asked how often they send e-mail to friends. Their responses are summarized in the following (incomplete) cross-tabulation. Separate tabulations are provided for social networkers (teens who have a profile on a social network site such as Facebook or Instagram) and for teens who are not social networkers. Fill in the empty cells in the following cross-tabulation:
Let S = the event the Internet-using teen is a social networker,
N = the event the Internet-using teen is not a social networker,
E = the event the Internet-using teen sends e-mail to friends every day, and
O = the event the Internet-using teen does not send e-mail to friends every day.
Consider this experiment: An Internet-using teenager is randomly selected and surveyed. The teen is categorized as a social networker or not a social networker and as someone who sends e-mail to friends every day or someone who does not.
22. What is the probability that the randomly selected Internet-using teenager does not send e-mail to friends every day?
23. What is the probability that the randomly selected Internet-using teenager is a non-social networker who does not send e-mail to friends every day?
What is the probability that a randomly selected Internet-using teenager does not send e-mail to friends every day given that he or she is a non-social networker?