Designing Studies Checkpoint 1 and 2

A person trying to gain access to a bank vault must pass through a series of three security doors. If an attempt to pass through a door is a failure, then the person will not make any further attempts. Let P denote a successful pass and F denote a failed pass. What is the sample space for this random experiment? S = {P, PP, PPP} S = {F, PF, PPF, PPP} S= {F, PF, PPF} S ={PPP, FPP, PFP, PPF, FFP, FPF, PFF, FFF} S = {F, FP, FFP, FFF}

S = {F, PF, PPF, PPP}

A fair four-sided die, where each face is represented by a different digit between 1 and 4, is rolled 5 times. What is the probability of rolling 5 ones in a row? (1/4)5 (1/4) * 5 (3/4)5 5 * (3/4) * (1/4)4 5 * (1/4) * (3/4)4 Feedback Correct. P(all 1s) = P(1st roll is 1 and 2nd roll is a 1 and 3rd roll is a 1 and 4th roll is a 1 and 5th roll is a 1). Since the five rolls are independent of each other, P(all 1s) = P(1) * P(1) * P(1) * P(1) * P(1) = P(1)5.

(1/4)^5 Correct. P(all 1s) = P(1st roll is 1 and 2nd roll is a 1 and 3rd roll is a 1 and 4th roll is a 1 and 5th roll is a 1). Since the five rolls are independent of each other, P(all 1s) = P(1) * P(1) * P(1) * P(1) * P(1) = P(1)5.

Let A and B be two disjoint events such that P(A) = 0.20 and P(B) = 0.60. What is P(A and B)? 0 0.12 0.68 0.80

0 Feedback Correct. If two events are disjoint, then by definition, P(A and B) = 0 (the two events cannot happen together).

In the last round of a chess tournament the final match is between Alice and Diego. The winner is the first player to win three games [sometimes called "best of 5"]. Assume that they are equally matched, so that each player has an equal probability of winning each game. What is the probability that the match will be finished after the first 3 games are played? 0.50 0.25 0.20 0.125

0.25

An engineering school reports that 55% of its students were male (M), 40% of its students were between the ages of 18 and 20 (A), and that 25% were both male and between the ages of 18 and 20. What is the probability of a random student being a female who is not between the ages of 18 and 20? 0.27 0.25 0.30 0.45

0.30 Feedback Correct. Use the Complement Rule with the General Addition Rule to get P(F and not A) = 1 − (P(M) + P(A) − P(M and A)), which equals 1 − (0.55 + 0.40 − 0.25) = 1 − 0.70 = 0.30.

A True/False quiz has three questions. When guessing, the probability of getting a question correct is the same as the probability of getting a question wrong. What is the probability that a student that guesses gets at least 2 questions correct? (Give your answer to 2 decimal places)

0.5 Correct. There are 8 equally likely outcomes in this sample space. Four of the outcomes (CCW, CWC, WCC, CCC) have at least 2 questions correct. So the probability is 4/8 = 0.5 that at least 2 questions are guessed correctly.

In a certain liberal arts college with about 10,000 students, 40% are males. If two students from this college are selected at random, what is the probability that they are of the same gender? 0.96 0.52 0.48 0.36 0.16

0.52 Feedback Correct. P(both of the same gender) = P(2 males or 2 females) = [disjoint events] P(2 males) + P(2 females) = [random selection → independent] (0.40 * 0.40) + (0.60 * 0.60) = 0.16 + 0.36 = 0.52.

A family plans to have 3 children. For each birth, assume that the probability of a boy is the same as the probability of a girl. What is the probability that they will have at least one boy and at least one girl? 0.5 0.125 0.75 0.875

0.75 Correct. The outcomes are equally likely, so the easiest way to work this problem is to write out the 8 outcomes in this sample space. In two outcomes the gender of all three children is the same (GGG, BBB). The other 6 outcomes contain at least one boy and one girl.

For safety reasons, four different alarm systems were installed in the vault containing the safety deposit boxes at a Beverly Hills bank. Each of the four systems detects theft with a probability of 0.99 independently of the others. The bank, obviously, is interested in the probability that when a theft occurs, at least one of the four systems will detect it. What is the probability that when a theft occurs, at least one of the four systems will detect it? (0.99)4 (0.01)4 1 − (0.99)4 1 − (0.01)4

1 − (0.01)4 Feedback Correct. We want to find the probability that "at least one system detects the theft," which is the complement of "none detects." P(at least one detects) = 1 − P(none detects) = 1 − (0.01)4.

For safety reasons, three different alarm systems were installed on the property of a famous movie star. Each of the three systems detects when a trespass occurs with a probability of 0.98 independently of the others. The movie star, obviously, is interested in the probability that when a trespass occurs, at least one of the three systems will detect it. What is the probability that when a trespass occurs, at least one of the systems will detect it? (0.98)3 (0.02)3 1 − (0.98)3 1 − (0.02)3 3 * (0.98)1(0.02)2

1 − (0.02)3 Correct. We want to find the probability that "at least one system detects the trespass," which is the complement of "none detects." P(at least one detects) = 1 − P(none detects) = 1 − (0.01)4.

A six-sided cube is rolled. What is the probability that the number is odd or less than 4? Event A: Numbers on a six-sided cube are odd: 1, 3, 5 Event B: Numbers on a six sided cube are less than 4: 1, 2, 3 1/2 2/3 5/6 1

2/3 Feedback Correct. P(A) = 3/6, P(B) = 3/6, P(A and B) = 2/6. We need to find P(A or B). Using the General Addition Rule: P(A or B) = P(A) + P(B) − P(A and B) = 3/6 + 3/6 − 2/6 = 4/6 = 2/3. Notice in this problem that the number 3 appears in both event A and event B. If we did not subtract the P(A and B), the answer would be 1, which we know is not true because the number 4 appears in neither event.

A True/False quiz has three questions. When guessing, the probability of getting a question correct is the same as the probability of getting a question wrong. What is the probability that a student that guesses gets at least 2 questions correct? (Give your answer to 2 decimal places)

4/8

According to the information that comes with a certain prescription drug, when taking this drug, there is a 20% chance of experiencing nausea (N) and a 50% chance of experiencing decreased sexual drive (D). The information also states that there is a 15% chance of experiencing both side effects. What formula will give you the probability of experiencing neither of these side effects? P("not N" and "not D") 1 − P("not N" and "not D") P(N or D) P(N and D) 1 − P(N and D)

P("not N" and "not D")

A person in a casino decides to play blackjack until he wins a game, but he will not play more than 3 games. Let W denote a win and L denote a loss. What is the sample space for this random experiment? S = {W, WL, WWL, WWW} S = {WWW, WWL, WLW, , WLL, LWW, LWL, LLW, LLL} S = {W, LW, LLW, LLL} S = {W, WW, WWW} S = {W, LW, LLW}

S = {W, LW, LLW, LLL}

A fair coin is tossed 12 times. Which of the following outcomes (i, ii, iii, or iv) is most likely? (i) H T H T H T H T H T H T (ii) H T T H H T T H T H H T (iii) H H H H H H H H H H H H (iv) T T T H T H H H H T H H (i) because there are an equal number of heads and tails. (ii) because there are an equal number of heads and tails but in a random order (iii) because heads are just as likely as tails (iv) because you won't necessarily get the same number of heads and tails with a fair coin (v)They are all equally likely.

They are all equally likely. Feedback Correct. Each of the sequences is equally likely with a probability of (1/2)12 (since the 12 tosses are independent)

In 2012, researchers working with a very large population of health records found that 9.3% of all Americans had diabetes (source: National Diabetes Statistics Report, 2014). Suppose a medical researcher randomly selects two individuals from a large population. Let A represent the event "the first individual has diabetes." Let B represent the event "the second individual has diabetes." True or false? A and B are independent events. True False

True

A couple decides to have three children. Let A define the event that the couple has at least 1 girl. What are the possible outcomes for this event? (G=girl, B=boy) {G, BG, BBG} {G, GG, GGG} {BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG} {GGG, GGB, GBG, BGG, GBB, BGB, BBG} {GBB, BGB, BBG} Feedback Correct. Event A is all outcomes in the sample space except BBB.

{GGG, GGB, GBG, BGG, GBB, BGB, BBG} Correct. Event A is all outcomes in the sample space except BBB.

A person must enter a 4 digit code to gain access to his cell phone. He will enter codes until he is successful, however he cannot try more than 3 times or the phone will lock him out. Let S denote a successful attempt and F denote a failed attempt. What is the sample space for this random experiment? {SSS, SSF, SFS, FSS, SFF, FSF, FFS, FFF} {S, FS, FFS} {S, FS, FFS, FFF} {S, SS, SSS} {S, SF, SSF, SSS}

{S, FS, FFS, FFF} Correct. The person stops trying when he successfully enters the code or when he has failed at all 3 attempts.

A person in a casino decides to play 3 games of blackjack. Let W denote a win and L denote a loss. Define the event A as "the person wins at least one game of blackjack." What are the possible outcomes for this event? {W, LW, LLW} {WWW, WWL, WLW, WLL, LWW, LWL, LLW} {WWL, LWL, LLW} {WWW, WWL, WLW, WLL, LWW, LWL, LLW, LLL} {W, WW, WWW} Feedback Correct. Event A is all outcomes in the sample space except LLL.

{WWW, WWL, WLW, WLL, LWW, LWL, LLW} Correct. Event A is all outcomes in the sample space except LLL.

A person in a casino decides to play 3 games of blackjack. Let W denote a win and L denote a loss. Define the event A as "the person wins at least one game of blackjack." What are the possible outcomes for this event? {WWW, WWL, WLW, WLL, LWW, LWL, LLW, LLL} {W, LW, LLW} {WWL, LWL, LLW} {WWW, WWL, WLW, WLL, LWW, LWL, LLW} {W, WW, WWW}

{WWW, WWL, WLW, WLL, LWW, LWL, LLW} Correct. Event A is all outcomes in the sample space except LLL.

For a criminal trial, 8 active and 4 alternate jurors are selected. Two of the alternate jurors are male and two are female. During the trial, two of the active jurors are dismissed. The judge decides to randomly select two replacement jurors from the 4 available alternates. What is the probability that both jurors selected are female? ¼ 1/12 ⅙ ½

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Four students attempt to register online at the same time for an Introductory Statistics class that is full. Two are freshmen and two are sophomores. They are put on a wait list. Prior to the start of the semester, two enrolled students drop the course, so the professor decides to randomly select two of the four wait list students and gives them a seat in the class. What is the probability that both students selected are freshmen? ⅙ ½ 1/12 ¼

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