Question 1

Simplify: 
A) 500
B) 501
C)  1 / 500 
D)  1 / 501 
E) None of the above.

Accepted Answer

The answer of Simplify: 
A) 500
B) 501
C)  1 /...

Question 2

A fair coin is tossed in the air ten times. What is the probability that it will land heads exactly nine times? Give your answer as a fraction in reduced form.
A) 1023/1024
B) 9/10
C) 5/512
D) 3/5
E) None of the above.

Accepted Answer

The probability of getting heads in a single toss is 1/2. The probability of getting exactly 9 heads in 10 tosses is given by the binomial probability formula: C(10,9) * (1/2)^9 * (1/2)^1 = 10 * (1/2)^10 = 10/1024 = 5/512.

Question 3

You toss a coin in the air six times and record the sequence of heads (H) and tails (T). How many outcomes would be in the sample space?
A) 2
B) 8
C) 16
D) 64
E) None of the above.

Accepted Answer

Each toss of a coin has 2 possible outcomes (Head or Tail). For 6 tosses, the number of outcomes in the sample space is $2^6 = 64$.

Question 4

A fair coin is tossed in the air ten times. What is the probability that it will land heads exactly one time? Give your answer as a fraction in reduced form.
A) 5/512
B) 1/2
C) 1023/1024
D) 1/10
E) None of the above.

Accepted Answer

The probability of getting heads on a single toss is 1/2. The probability of getting tails on a single toss is also 1/2. The probability of getting heads exactly one time in ten tosses is (1/2)^1 * (1/2)^9 * 10 = 5/512.

Question 5

You have 20 vintage concert t-shirts. Sadly your wife is making you throw away all but five of the shirts. How many different groups of shirts can you choose?
A) 56
B) 105
C) 15504
D) 1860480
E) None of the above.

Accepted Answer

The number of different groups of shirts you can choose is given by the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, and k is the number of items to choose. Here, n = 20 and k = 5, so C(20, 5) = 20! / (5! * 15!) = 15504.

Question 6

You toss two fair six-sided dice and two coins into the air at the same time and record the outcome when they land. What is the probability of the following event? Give your answer as a fraction in
Reduced form.
E1 :"both dice are different and both coins are the same"
A) 1/12
B) 5/12
C) 1/4
D) 11/36
E) None of the above.

Accepted Answer

The answer of You toss two fair six-sided dice and...

Question 7

You are choosing 20 Facebook friends to participate in a Flash-Mod. If you have 850 Facebook friends, then how many different groups of 20 Facebook friends can you choose?
A) 
B) 
C)  20 ! 
D)  850 ! 
E) None of the above.

Accepted Answer

The answer of You are choosing 20 Facebook friends to...

Question 8

On Valentine's Day you and your date attend a fancy dinner where the following fixed price menu is being served. For $32 per person, your meal includes a choice of appetizer, salad, entrée, and desert. If there are 4 different appetizers, 6 different salads, 8 different entrées, and 3 different deserts to Choose from, then how many different dinner choices are possible? (Note: Assume that you must Choose exactly one from each category.)
A) 21
B) 576
C) 1064
D) 41824
E) None of the above.

Accepted Answer

To find the total number of dinner choices possible, multiply the number of options in each category: 4 appetizers * 6 salads * 8 entrées * 3 desserts = 576 different dinner choices.

Question 9

A scratch-off lottery ticket costs $5.00 to purchase, and it pays out a prize of $0, $5, $10 and $100 with probabilities of 90%, 8%, 1.5% and 0.5%, respectively. What is the expected value of the profit of this Scratch-off lottery ticket?
A) -$4.49
B) -$4.25
C) -$4.08
D) -$3.95
E) None of the above.

Accepted Answer

The answer of A scratch-off lottery ticket costs $5.00 to...

Question 10

There are 50 contestants in the Miss America pageant. The judges need to pick 1^st place, 2^nd place, and 3^rd place. How many different ways can the judges choose the arrangement of winners? A) 19600 B) 125000 C) 15253 D) 117600 E) None of the above.

Accepted Answer

The number of different ways to choose the arrangement of winners is calculated using permutations since the order matters. For the first place, there are 50 options, for the second place, there are 49 (since one contestant has already been chosen), and for the third place, there are 48 options. Thus, the total number of arrangements is 50 * 49 * 48 = 117,600.

Question 11

The odds that it will rain tomorrow are 3 to 5. What is the probability that it will rain tomorrow?
(a) 25%
(b) 35%
(c) 60%
(d) 80%
(e) None of the above.

Accepted Answer

The answer of The odds that it will rain tomorrow...

Question 12

A computer password is composed of five characters. If each character must be a letter (A - Z), then how many different passwords can be created which use different vowels for the first three characters And different consonants for the last two characters? (Note: The vowels are A, E, I, O, and U.)
A) 253
B) 7542
C) 25200
D) 589920
E) None of the above.

Accepted Answer

There are 5 vowels and we need to choose 3 different ones for the first three characters, which can be done in 5P3 = 60 ways. There are 21 consonants and we need to choose 2 different ones for the last two characters, which can be done in 21P2 = 420 ways. Therefore, the total number of different passwords is 60 * 420 = 25200.

Question 13

You shoot ten free throws and record the number that you successfully make. How many possible outcomes would be in the sample space?
A) 1024
B) 1023
C) 10
D) 11
E) None of the above.

Accepted Answer

For each free throw, there are two possible outcomes: make or miss. However, the question asks for the number of successful shots, which can range from 0 to 10, inclusive. This gives us a total of 11 possible outcomes.

Question 14

You toss two fair six-sided dice and two coins into the air at the same time and record the outcome when they land. What is the probability of the following event? Give your answer as a fraction in
Reduced form.
E1 :"both dice are the same and both coins are the same"
A) 1/9
B) 13/36
C) 5/18
D) 1/12
E) None of the above.

Accepted Answer

The probability that both dice show the same number is 1/6 (since there are 6 possible outcomes for a die and only 1 favorable outcome for both being the same). The probability that both coins show the same side is 1/2 (since there are 2 possible outcomes for a coin and only 1 favorable outcome for both being the same). Therefore, the probability of both events happening simultaneously is (1/6) * (1/2) = 1/12.

Question 15

The chance that it will rain tomorrow is 30%. Thus, the odds that it will rain tomorrow are what? 
A) 30 to 100
B) 3 to 7
C) 30 to 1
D) 5 to 6
E) None of the above.

Accepted Answer

Odds are calculated as the ratio of the probability of the event happening to the probability of the event not happening. With a 30% chance of rain, there's a 70% chance it won't rain. Thus, the odds are 30:70, which simplifies to 3:7.

Question 16

On Valentine's Day you and your date attend a fancy dinner where the following fixed price menu is being served. For $32 per person the meal includes your choice of appetizer, salad, entrée, and desert. If there are 5 different appetizers, 3 different salads, 7 different entrées, and 4 different deserts to Choose from, then how many different dinner choices are possible? (Note: Assume that you must Choose exactly one from each category.)
A) 350
B) 420
C) 684
D) 1425
E) None of the above.

Accepted Answer

To find the total number of dinner choices possible, multiply the number of options in each category: 5 appetizers * 3 salads * 7 entrées * 4 desserts = 420.

Question 17

You have 800,000 people who follow you on Twitter. You would like to choose a president, a vicepresident and a secretary from your 800,000 followers to form a Fan-Club. How many different ways could this be done, realizing that order of the choices will matter?
A) 
B) 
C)  800000 ! 
D)  3 ! 
E) None of the above.

Accepted Answer

The answer of You have 800,000 people who follow you...

Question 18

Simplify: 
A) 501/500
B)  500 / 499 
C)  500 / 501 
D)  499 / 500 
E) None of the above.

Accepted Answer

The answer of Simplify: 
A) 501/500
B)  500 / 499...

Question 19

A scratch-off lottery ticket costs $1.00 to purchase, and it pays out a prize of $0, $1, $5 and $10 with probabilities of 85%, 12%, 2% and 1%, respectively. What is the expected value of the profit of this Scratch-off lottery ticket?
A) -$0.88
B) -$0.73
C) -$0.68
D) -$0.56
E) None of the above.

Accepted Answer

The answer of A scratch-off lottery ticket costs $1.00 to...

Question 20

A computer password is composed of seven characters. If each character must be a letter (A - Z), then how many different passwords can be constructed so that the password is the same both forwards and Backwards? (Note: Examples would be ETTFTTE or RACECAR.)
A) 245712
B) 358800
C) 456976
D) 5866254125
E) None of the above.

Accepted Answer

A palindrome password of 7 characters has the form ABCDCBA. There are 26 choices for each of the first four characters (A, B, C, D), resulting in 26^4 = 456976 possible passwords.

Quiz 16: Probabilities, Odds, and Expectations: Measuring Uncertainty and Risk

Simplify:

A fair coin is tossed in the air ten times. What is the probability that it will land heads exactly nine times? Give your answer as a fraction in reduced form.

You toss a coin in the air six times and record the sequence of heads (H) and tails (T) . How many outcomes would be in the sample space?

A fair coin is tossed in the air ten times. What is the probability that it will land heads exactly one time? Give your answer as a fraction in reduced form.

You have 20 vintage concert t-shirts. Sadly your wife is making you throw away all but five of the shirts. How many different groups of shirts can you choose?

You toss two fair six-sided dice and two coins into the air at the same time and record the outcome when they land. What is the probability of the following event? Give your answer as a fraction in Reduced form. E1 :"both dice are different and both coins are the same"

You are choosing 20 Facebook friends to participate in a Flash-Mod. If you have 850 Facebook friends, then how many different groups of 20 Facebook friends can you choose?

A scratch-off lottery ticket costs $5.00 to purchase, and it pays out a prize of $0, $5, $10 and $100 with probabilities of 90%, 8%, 1.5% and 0.5%, respectively. What is the expected value of the profit of this Scratch-off lottery ticket?

There are 50 contestants in the Miss America pageant. The judges need to pick 1^st place, 2^nd place, and 3^rd place. How many different ways can the judges choose the arrangement of winners?

The odds that it will rain tomorrow are 3 to 5. What is the probability that it will rain tomorrow? (a) 25% (b) 35% (c) 60% (d) 80% (e) None of the above.

A computer password is composed of five characters. If each character must be a letter (A - Z) , then how many different passwords can be created which use different vowels for the first three characters And different consonants for the last two characters? (Note: The vowels are A, E, I, O, and U.)

You shoot ten free throws and record the number that you successfully make. How many possible outcomes would be in the sample space?

You toss two fair six-sided dice and two coins into the air at the same time and record the outcome when they land. What is the probability of the following event? Give your answer as a fraction in Reduced form. E1 :"both dice are the same and both coins are the same"

The chance that it will rain tomorrow is 30%. Thus, the odds that it will rain tomorrow are what?

You have 800,000 people who follow you on Twitter. You would like to choose a president, a vicepresident and a secretary from your 800,000 followers to form a Fan-Club. How many different ways could this be done, realizing that order of the choices will matter?

Simplify:

A scratch-off lottery ticket costs $1.00 to purchase, and it pays out a prize of $0, $1, $5 and $10 with probabilities of 85%, 12%, 2% and 1%, respectively. What is the expected value of the profit of this Scratch-off lottery ticket?

A computer password is composed of seven characters. If each character must be a letter (A - Z) , then how many different passwords can be constructed so that the password is the same both forwards and Backwards? (Note: Examples would be ETTFTTE or RACECAR.)

Quiz 16: Probabilities, Odds, and Expectations: Measuring Uncertainty and Risk

Simplify:

A fair coin is tossed in the air ten times. What is the probability that it will land heads exactly nine times? Give your answer as a fraction in reduced form.

You toss a coin in the air six times and record the sequence of heads (H) and tails (T) . How many outcomes would be in the sample space?

A fair coin is tossed in the air ten times. What is the probability that it will land heads exactly one time? Give your answer as a fraction in reduced form.

You have 20 vintage concert t-shirts. Sadly your wife is making you throw away all but five of the shirts. How many different groups of shirts can you choose?

You toss two fair six-sided dice and two coins into the air at the same time and record the outcome when they land. What is the probability of the following event? Give your answer as a fraction in Reduced form. E1 :"both dice are different and both coins are the same"

You are choosing 20 Facebook friends to participate in a Flash-Mod. If you have 850 Facebook friends, then how many different groups of 20 Facebook friends can you choose?

A scratch-off lottery ticket costs $5.00 to purchase, and it pays out a prize of $0, $5, $10 and $100 with probabilities of 90%, 8%, 1.5% and 0.5%, respectively. What is the expected value of the profit of this Scratch-off lottery ticket?

There are 50 contestants in the Miss America pageant. The judges need to pick 1st place, 2nd place, and 3rd place. How many different ways can the judges choose the arrangement of winners?

The odds that it will rain tomorrow are 3 to 5. What is the probability that it will rain tomorrow? (a) 25% (b) 35% (c) 60% (d) 80% (e) None of the above.

A computer password is composed of five characters. If each character must be a letter (A - Z) , then how many different passwords can be created which use different vowels for the first three characters And different consonants for the last two characters? (Note: The vowels are A, E, I, O, and U.)

You shoot ten free throws and record the number that you successfully make. How many possible outcomes would be in the sample space?

You toss two fair six-sided dice and two coins into the air at the same time and record the outcome when they land. What is the probability of the following event? Give your answer as a fraction in Reduced form. E1 :"both dice are the same and both coins are the same"

The chance that it will rain tomorrow is 30%. Thus, the odds that it will rain tomorrow are what?

You have 800,000 people who follow you on Twitter. You would like to choose a president, a vicepresident and a secretary from your 800,000 followers to form a Fan-Club. How many different ways could this be done, realizing that order of the choices will matter?

Simplify:

A scratch-off lottery ticket costs $1.00 to purchase, and it pays out a prize of $0, $1, $5 and $10 with probabilities of 85%, 12%, 2% and 1%, respectively. What is the expected value of the profit of this Scratch-off lottery ticket?

A computer password is composed of seven characters. If each character must be a letter (A - Z) , then how many different passwords can be constructed so that the password is the same both forwards and Backwards? (Note: Examples would be ETTFTTE or RACECAR.)

There are 50 contestants in the Miss America pageant. The judges need to pick 1^st place, 2^nd place, and 3^rd place. How many different ways can the judges choose the arrangement of winners?