Question 1

i. A probability is usually expressed as a decimal, such as 0.70 or 0.27, but it may be given as a fraction, such as 7/10 or 27/100. ii. The probability of 1 represents something that is certain to happen.
iii. The probability of 0 represents something that cannot happen.

Accepted Answer

All three statements are true and commonly accepted definitions of probabilities. The first statement describes the common representations of probabilities, while the second and third statements give the extreme values that probabilities can take (certain and impossible, respectively).

Question 2

i. The Cunard luxury liner, Queen Elizabeth 2, cannot be docked in Hong Kong and Bangkok at the same time. Events such as these that cannot occur simultaneously are said to be collectively exhaustive. ii. If there are 'm' ways of doing one thing and 'n' ways of doing another thing, the multiplication formula states that there are (m)(n) ways of doing both.
iii. A permutation is an arrangement of a set of objects in which there is an order from the first through the last.

Accepted Answer

- Statement (i) incorrectly defines "collectively exhaustive." Events that cannot occur simultaneously are actually "mutually exclusive," not collectively exhaustive. Collectively exhaustive events cover all possible outcomes.- Statement (ii) is correct. The multiplication principle states that if there are m ways to do one thing and n ways to do another, then there are m*n ways to do both.- Statement (iii) correctly defines a permutation as an ordered arrangement of objects.

Question 3

A sales representative calls on four hospitals in York Region. It is immaterial what order he calls on them. How many ways can he organize his calls?

Accepted Answer

Since the order in which the sales representative calls on the hospitals does not matter, we are looking for the number of combinations, not permutations. However, since he will visit all four hospitals, there is actually only one way to choose all four hospitals out of four, which might seem confusing given the question's phrasing. The correct interpretation, considering the context of organizing or scheduling visits, implies the order of visits matters (despite the initial statement), turning this into a permutations problem. Therefore, the number of ways to organize his calls is 4! (4 factorial), which is 4 x 3 x 2 x 1 = 24.

Question 4

i. A probability is usually expressed as a decimal, such as 0.70 or 0.27, but it may be given as a fraction, such as 7/10 or 27/100. ii. The closer a probability is to 0, the more likely that an event will happen.
iii. The closer the probability is to 1.00, the more likely an event will not happen.

Accepted Answer

Explanation: Statement (i) is correct as probabilities can indeed be expressed as decimals or fractions. However, statements (ii) and (iii) are incorrect because the closer a probability is to 0, the less likely an event is to happen, and the closer a probability is to 1.00, the more likely an event is to happen.

Question 5

Consideration is being given to forming a Super Ten Basketball Conference. The top 10 university basketball teams in the country, based on past records, would be members of the Super Ten Conference. Each team would play every other team in the conference during the season and the team winning the most games would be declared the national champion. How many games would the conference commissioner have to schedule each year? (Remember, McMaster versus Alberta is the same as Alberta versus McMaster.)

Accepted Answer

The total number of games can be calculated using the formula for combinations since each game is between two teams and the order does not matter. The formula for combinations is C(n, k) = n! / [k!(n-k)!], where n is the total number of teams and k is the number of teams in each game (2 in this case). So, for 10 teams, it would be C(10, 2) = 10! / [2!(10-2)!] = 45 games.

Question 6

A rug manufacturer has decided to use 7 compatible colours in her rugs. However, in weaving a rug, only 5 spindles can be used. In advertising, the rug manufacturer wants to indicate the number of different colour groupings for sale. How many colour groupings using the seven colours taken five at a time are there? (This assumes that 5 different colours will go into each rug, i.e., there are no repetitions of colour.)

Accepted Answer

The number of different colour groupings can be found by using the combination formula, which is: 
nCr = n! / r!(n-r)!
where n is the total number of colours (7) and r is the number of colours taken at a time (5). 
Plugging in the values, we get: 
7C5 = 7! / 5!(7-5)! 
= 7! / (5!2!) 
= (7x6x5x4x3x2x1) / (5x4x3x2x1x2x1) 
= 21x7 
= 147

Therefore, there are 2,520 different colour groupings using the seven colours taken five at a time.

Question 7

i. The Cunard luxury liner, Queen Elizabeth 2, cannot be docked in Hong Kong and Bangkok at the same time. Events such as these that cannot occur simultaneously are said to be mutually exclusive. ii. If there are 'm' ways of doing one thing and 'n' ways of doing another thing, the multiplication formula states that there are (m) $\bullet$ (n) ways of doing both.
iii. A permutation is an arrangement of a set of objects in which order does not matter.

Accepted Answer

Statement (i) is true as events that cannot occur simultaneously are indeed mutually exclusive. Statement (ii) is also true as per the multiplication formula which states that there are (m) ways of doing one thing and (n) ways of doing another thing make a total of (m) x (n) ways of doing both. However, statement (iii) is false as a permutation is an arrangement of a set of objects in which order does matter. Hence, option B is the best answer.

Question 8

You have the assignment of designing colour codes for different parts. Three colours are to be used on each part, but a combination of three colours used for one part cannot be rearranged and used to identify a different part. This means that if green, yellow and violet were used to identify a camshaft, yellow, violet and green (or any other combination of these three colours) could not be used to identify a pinion gear. If there are 35 combinations, how many colours were available?

Accepted Answer

Since each part can be identified using a unique combination of 3 colours, we can treat each combination as a distinct element. There are 35 such elements, which is equivalent to the number of ways of choosing 3 elements out of n. This can be expressed as nC3 = 35, which gives n(n-1)(n-2)/6 = 35. Solving for n, we get n = 7. Therefore, there were 7 colours available.

Question 9

i. A probability is usually expressed as a decimal, such as 0.70 or 0.27, but it may be given as a fraction, such as 7/10 or 27/100. ii. The closer a probability is to 0, the more likely that an event will not happen.
iii. The closer the probability is to 1.00, the more likely an event will happen.

Accepted Answer

All three statements are correct and provide accurate information regarding probabilities. The first statement describes how probabilities are typically expressed, either as decimals or fractions. The second statement correctly states that as the probability gets closer to 0, the event is less likely to happen. The third statement correctly states that as the probability gets closer to 1, the event is more likely to happen.

Question 10

An electronics firm manufactures three models of stereo receivers, two cassette decks, four speakers and three CD players. When the four types of components are sold together, they form a "system." How many different systems can the electronic firm offer?

Accepted Answer

To find the number of different systems the electronics firm can offer, we need to multiply the number of options for each component together. Thus, the number of different systems is: 3 (stereo receivers) x 2 (cassette decks) x 4 (speakers) x 3 (CD players) = 72. Therefore, the best choice is C.

Question 11

The numbers 0 through 9 are to be used in code groups of four to identify an item of clothing. Code 1083 might identify a blue blouse, size medium. The code group 2031 might identify a pair of pants, size 18, and so on. Repetitions of numbers are not permitted, i.e., the same number cannot be used more than once in a total sequence. As examples, 2256, 2562 or 5559 would not be permitted. How many different code groups can be designed?

Accepted Answer

Since repetitions are not allowed and each code group consists of 4 unique digits selected from 0 through 9 (10 options), the number of different code groups can be calculated using the permutation formula for selecting 4 unique digits out of 10, which is 10P4 = 10! / (10-4)! = 10*9*8*7 = 5,040.

Question 12

i. The probability of an event, based on a classical approach, is defined as the number of favourable outcomes divided by the total number of possible outcomes. ii. If among several events only one can occur at a time, we refer to these events as being mutually exclusive events.
iii. The probability of rolling a 3 or 2 on a single die is an example of conditional probability.

Accepted Answer

Statement (i) is correct and is a fundamental concept in classical probability. Statement (ii) is also correct and applies to situations where events cannot occur simultaneously. However, statement (iii) is incorrect as the probability of rolling a 3 or 2 on a single die is an example of simple probability, not conditional probability. Conditional probability involves the probability of an event happening given that another event has already occurred.

Question 13

Six basic colours are to be used in decorating a new condominium. They are to be applied to a unit in groups of four colours. One unit might have gold as the principal colour, blue as a complementary colour, red as the accent colour and touches of white. Another unit might have blue as the principal colour, white as the complimentary colour, gold as the accent colour and touches of red. If repetitions are not permitted, how many different units can be decorated?

Accepted Answer

The problem is a permutation problem because the order in which the colors are used (principal, complementary, accent, touches) matters and repetitions are not allowed. With 6 colors to choose from, and choosing 4 for each unit, the number of different units can be calculated using the permutation formula P(n, r) = n! / (n-r)!, where n is the total number of items to pick from, r is the number of items to pick, ! denotes factorial, which is the product of all positive integers up to that number. Here, n=6 (total colors) and r=4 (colors chosen for each unit), so P(6, 4) = 6! / (6-4)! = 6! / 2! = (6*5*4*3) / (2*1) = 360.

Question 14

i. An illustration of an experiment is turning the ignition key of an automobile as it comes off the assembly line to determine whether or not the engine will start. ii. If there are 'm' ways of doing one thing and 'n' ways of doing another thing, the multiplication formula states that there are (m)*(n) ways of doing both.
iii. A permutation is an arrangement of a set of objects in which there is an order from the first through the last.

Accepted Answer

All three statements are correct. Statement i correctly describes an experiment, statement ii accurately explains the multiplication principle in combinatorics, and statement iii correctly defines a permutation.

Question 15

What does  equal?

Accepted Answer

The answer of What does  equal?
...

Question 16

A builder has agreed not to erect all "lookalike" homes in a new subdivision. Five exterior designs are offered to potential homebuyers. The builder has standardized three interior plans that can be incorporated in any of the five exteriors. How many different ways are the exterior and interior plans offered to potential homebuyers?

Accepted Answer

There are 5 exterior designs and 3 interior plans. So, the total number of different ways the exterior and interior plans are offered to potential homebuyers is 5 * 3 = 15.

Question 17

i. An experiment is an activity that is either observed or measured. ii. If an experiment, such as a die-tossing experiment, has a set of events that includes every possible outcome, the set of events is called collectively exhaustive.
iii. The combination formula is: n!/(n - r)!

Accepted Answer

The combination formula is correctly given by n! / [(n-r)! * r!], not n! / (n-r)!, making statement (iii) incorrect. Statements (i) and (ii) correctly describe an experiment and the concept of collectively exhaustive events, respectively.

Question 18

Six basic colours are to be used in decorating a new condominium. They are to be applied to a unit in groups of four colours. One unit might have gold as the principal colour, blue as a complementary colour, red as the accent colour and touches of white. Another unit might have blue as the principal colour, white as the complimentary colour, gold as the accent colour and touches of red. If repetitions are permitted, how many different units can be decorated?

Accepted Answer

There are 6 choices for the principal colour, and for each of those choices, there are 5 choices for the complementary colour (since it cannot be the same as the principal colour). For each of those 6*5 = 30 combinations, there are 4 choices for the accent colour. Finally, there are 6 choices for the touches of a fourth colour. Therefore, the total number of units that can be decorated is 6 x 5 x 4 x 6 = 1,296.

Question 19

There are two letters C and D. If repetitions such as CC are permitted, how many permutations are possible?

Accepted Answer

Since there are 2 letters (C and D) and repetitions are allowed, for each position in a permutation, there are 2 choices. For a 2-letter permutation, the total number of permutations is 2^2 = 4.

Question 20

i. A subjective probability can be assigned to an event by an individual based on the individual's knowledge about the event. ii. The probability that you would assign to the likelihood that the Hamilton Tiger Cats will be in the Grey Cup this season must be between 0 and 10.
iii. A probability is a number from -1 to +1 inclusive that measures one's belief that an event resulting from an experiment will occur.

Accepted Answer

The answer of i. A subjective probability can be assigned...

Quiz 4: A Survey of Probability Concepts

i. A probability is usually expressed as a decimal, such as 0.70 or 0.27, but it may be given as a fraction, such as 7/10 or 27/100. ii. The probability of 1 represents something that is certain to happen. iii. The probability of 0 represents something that cannot happen.

A sales representative calls on four hospitals in York Region. It is immaterial what order he calls on them. How many ways can he organize his calls?

i. A probability is usually expressed as a decimal, such as 0.70 or 0.27, but it may be given as a fraction, such as 7/10 or 27/100. ii. The closer a probability is to 0, the more likely that an event will happen. iii. The closer the probability is to 1.00, the more likely an event will not happen.

i. A probability is usually expressed as a decimal, such as 0.70 or 0.27, but it may be given as a fraction, such as 7/10 or 27/100. ii. The closer a probability is to 0, the more likely that an event will not happen. iii. The closer the probability is to 1.00, the more likely an event will happen.

An electronics firm manufactures three models of stereo receivers, two cassette decks, four speakers and three CD players. When the four types of components are sold together, they form a "system." How many different systems can the electronic firm offer?

What does equal?

i. An experiment is an activity that is either observed or measured. ii. If an experiment, such as a die-tossing experiment, has a set of events that includes every possible outcome, the set of events is called collectively exhaustive. iii. The combination formula is: n!/(n - r) !

There are two letters C and D. If repetitions such as CC are permitted, how many permutations are possible?