Probability

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Learning Objectives

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Learning Objectives

Understand the definition of theoretical (classical) probability.
Calculate the probability of an event based on favorable outcomes and total outcomes.
Differentiate between sure events, impossible events, and elementary events.
Recognize the relationship between complementary events and their probabilities.
Apply the concept of equally likely outcomes in various probability experiments.
Analyze and solve problems involving empirical and theoretical probabilities.
Identify common misconceptions related to probability calculations.

Revision Notes & Summary

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Probability Notes

Key Concepts

Theoretical Probability:
- Defined as:
  
  P(E) = $\frac{\text{Number of outcomes favourable to E}}{\text{Number of all possible outcomes}}$
- Assumes outcomes are equally likely.
Types of Events:
- Sure Event: Probability = 1
- Impossible Event: Probability = 0
- Elementary Event: An event with only one outcome.
- Complementary Events: For any event E, $P(E) + P(not E) = 1$
Probability Range:
- $0 \leq P(E) \leq 1$

Examples

Coin Toss:
- Outcomes: Head (H) or Tail (T)
- Probability of Head: $P(H) = \frac{1}{2}$
- Probability of Tail: $P(T) = \frac{1}{2}$
Die Roll:
- Outcomes: 1, 2, 3, 4, 5, 6
- Probability of rolling a 3: $P(3) = \frac{1}{6}$

Important Formulas

Probability of an Event:
$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total outcomes}}$
Complementary Probability:
$P(E) = 1 - P(not E)$

Common Mistakes

Misunderstanding equally likely outcomes. Not all experiments have equally likely outcomes (e.g., drawing from a bag with different colored balls).
Assuming probabilities can be negative or exceed 1.

Tips

Always check if outcomes are equally likely before applying theoretical probability.
Remember that the sum of probabilities of all possible outcomes must equal 1.

Exam Tips & Common Mistakes

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Common Mistakes and Exam Tips in Probability

Common Pitfalls

Misunderstanding Equally Likely Outcomes: Students often assume that all outcomes in an experiment are equally likely without verifying. For example, when drawing a ball from a bag containing different colored balls, the outcomes are not equally likely if the number of each color is different.
Incorrect Probability Calculation: Some students calculate probabilities based on the number of outcomes rather than the number of favorable outcomes. For instance, when calculating the probability of rolling a specific number on a die, they might mistakenly think there are more outcomes than there actually are.
Confusing Complementary Events: Students may forget that the sum of the probabilities of an event and its complement equals 1. For example, if the probability of an event E is 0.3, the probability of not E should be calculated as 1 - 0.3 = 0.7.
Assuming Independence Incorrectly: In problems involving multiple events, students sometimes incorrectly assume that events are independent when they are not. For example, drawing cards from a deck without replacement affects the probabilities of subsequent draws.

Tips for Success

Always Define Your Sample Space: Clearly outline all possible outcomes before calculating probabilities. This helps in identifying favorable outcomes accurately.
Check for Equally Likely Outcomes: Before applying the formula for probability, ensure that the outcomes are indeed equally likely. If not, adjust your calculations accordingly.
Use Complementary Events: If calculating the probability of an event seems complex, consider using the complementary event to simplify your calculations.
Practice with Real-Life Examples: Engage with practical examples, such as games of chance or everyday decisions, to better understand probability concepts.
Review Common Probability Formulas: Familiarize yourself with key probability formulas and definitions, such as the probability of an event being the number of favorable outcomes divided by the total number of outcomes.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

\frac{1}{4}

\frac{1}{2}

\frac{1}{3}

\frac{1}{13}

Correct Answer: A

Solution:

There are 13 spades in a deck of 52 cards. Therefore, the probability of drawing a spade is

\frac{13}{52} = \frac{1}{4}

0.375

0.5

0.625

0.25

Correct Answer: A

Solution:

The probability of drawing a red ball is given by the ratio of the number of red balls to the total number of balls. Thus,

P(\text{red}) = \frac{3}{3 + 5} = \frac{3}{8} = 0.375

\frac{1}{8}

\frac{3}{8}

\frac{1}{2}

\frac{5}{8}

Correct Answer: B

Solution:

The possible outcomes when a coin is tossed three times are HHH, HHT, HTH, THH, TTH, THT, HTT, TTT. The number of outcomes with exactly two heads is 3 (HHT, HTH, THH). Therefore, the probability is

\frac{3}{8}

0.25

0.5

0.75

Correct Answer: B

Solution:

The probability of getting a tail when a coin is tossed is

P(\text{tail}) = \frac{1}{2} = 0.5

\frac{3}{13}

\frac{1}{4}

\frac{1}{13}

\frac{9}{52}

Correct Answer: A

Solution:

There are 12 face cards in a deck (3 face cards per suit and 4 suits). Therefore, the probability is

\frac{12}{52} = \frac{3}{13}

0.231

0.25

0.192

0.308

Correct Answer: A

Solution:

There are 12 face cards in a deck of 52 cards. Thus, the probability is

P(\text{face card}) = \frac{12}{52} = \frac{3}{13} \approx 0.231

0.25

0.5

0.75

Correct Answer: B

Solution:

There are two possible outcomes when a coin is tossed: head or tail. The probability of getting a tail is

P(\text{tail}) = \frac{1}{2} = 0.5

\frac{1}{3}

\frac{1}{2}

\frac{1}{6}

\frac{2}{3}

Correct Answer: A

Solution:

The die has 6 faces, with the letter 'A' appearing on 2 of them. Therefore, the probability of getting 'A' is

\frac{2}{6} = \frac{1}{3}

1/4

1/2

3/4

1/3

Correct Answer: C

Solution:

The possible outcomes are HH, HT, TH, TT. The outcomes with at least one head are HH, HT, TH, which are 3 in total. Therefore, the probability is 3/4.

\frac{1}{6}

\frac{1}{3}

\frac{1}{2}

\frac{2}{3}

Correct Answer: B

Solution:

The numbers greater than 4 on a die are 5 and 6. There are 2 favorable outcomes out of 6 possible outcomes. Thus, P(number > 4) = \frac{2}{6} = \frac{1}{3}.

1/3

1/2

1/6

2/3

Correct Answer: A

Solution:

The numbers greater than 4 on a die are 5 and 6. Therefore, there are 2 favorable outcomes. The probability is 2/6 = 1/3.

0.25

0.5

0.75

1.0

Correct Answer: C

Solution:

The possible outcomes are HH, HT, TH, TT. The probability of getting at least one head is

\frac{3}{4} = 0.75

\frac{1}{4}

\frac{1}{8}

\frac{1}{2}

\frac{1}{3}

Correct Answer: A

Solution:

There are 2 outcomes for each toss: head (H) or tail (T). The possible outcomes for 3 tosses are HHH, HHT, HTH, THH, TTH, THT, HTT, TTT. Only HHH and TTT result in the same outcome. Thus, the probability is

\frac{2}{8} = \frac{1}{4}

\frac{1}{8}

\frac{1}{4}

\frac{1}{2}

\frac{3}{8}

Correct Answer: A

Solution:

The probability of getting heads on a single toss is \frac{1}{2}. Therefore, the probability of getting heads on all three tosses is \left(\frac{1}{2}\right)^3 = \frac{1}{8}.

1/2

1/3

1/6

2/3

Correct Answer: A

Solution:

The possible outcomes when a die is thrown are 1, 2, 3, 4, 5, 6. The even numbers are 2, 4, and 6. Therefore, the probability of getting an even number is the number of favorable outcomes (3) divided by the total number of outcomes (6), which is 1/2.

\frac{1}{8}

\frac{1}{2}

\frac{3}{8}

\frac{5}{8}

Correct Answer: B

Solution:

The odd numbers on the spinner are 1, 3, 5, and 7. There are 4 odd numbers out of 8 possible outcomes, so the probability is

\frac{4}{8} = \frac{1}{2}

1/4

1/2

1/8

3/8

Correct Answer: A

Solution:

The numbers greater than 6 are 7 and 8. Therefore, there are 2 favorable outcomes. The probability is 2/8 = 1/4.

1/3

3/9

3/10

1/9

Correct Answer: C

Solution:

The total number of balls is 4 + 3 + 2 = 9. The number of blue balls is 3. Therefore, the probability of drawing a blue ball is 3/9 = 1/3.

3/13

1/4

1/13

9/52

Correct Answer: A

Solution:

There are 12 face cards in a deck of 52 cards (4 Jacks, 4 Queens, 4 Kings). Therefore, the probability is 12/52 = 3/13.

1/8

3/8

1/2

1/4

Correct Answer: B

Solution:

The possible outcomes are HHT, HTH, THH. There are 3 favorable outcomes. Total possible outcomes are 8. So, the probability is

\frac{3}{8}

1/6

1/2

1/3

2/3

Correct Answer: B

Solution:

The possible outcomes when a die is thrown are 1, 2, 3, 4, 5, and 6. The numbers less than 4 are 1, 2, and 3. Therefore, the probability is

\frac{3}{6} = \frac{1}{2}

1/6

1/36

5/36

1/12

Correct Answer: A

Solution:

The possible outcomes when two dice are thrown are 36. The combinations that result in a sum of 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), which are 6 in total. Therefore, the probability is 6/36 = 1/6.

0.33

0.5

0.67

0.17

Correct Answer: A

Solution:

The numbers less than 3 on a die are 1 and 2. So, the probability is

P(\text{less than 3}) = \frac{2}{6} = \frac{1}{3} = 0.33

\frac{1}{3}

\frac{1}{2}

\frac{1}{6}

\frac{2}{3}

Correct Answer: B

Solution:

The die has 6 faces with letters A, B, C, D, E, and A. The probability of getting 'A' is the number of 'A' faces divided by the total number of faces. Since there are 2 'A' faces, the probability is

\frac{2}{6} = \frac{1}{3}

1/2

3/8

5/8

1/4

Correct Answer: B

Solution:

The prime numbers between 1 and 8 are 2, 3, 5, and 7. Thus, there are 4 prime numbers. The probability is 4/8 = 1/2.

\frac{1}{9}

\frac{1}{12}

\frac{1}{8}

\frac{1}{6}

Correct Answer: B

Solution:

To find the probability of the sum being 9, we list all possible outcomes: (3,6), (4,5), (5,4), (6,3). There are 4 favorable outcomes. Total possible outcomes when a die is thrown twice is 6 \times 6 = 36. Thus, the probability is \frac{4}{36} = \frac{1}{9}.

\frac{1}{8}

\frac{1}{4}

\frac{3}{8}

\frac{1}{2}

Correct Answer: B

Solution:

The multiples of 4 in the range 1 to 8 are 4 and 8. Therefore, there are 2 favorable outcomes. The probability is \frac{2}{8} = \frac{1}{4}.

\frac{1}{7}

\frac{1}{3}

\frac{1}{5}

\frac{1}{2}

Correct Answer: A

Solution:

The probability of drawing two blue marbles is \frac{\binom{6}{2}}{\binom{15}{2}} = \frac{15}{105} = \frac{1}{7}.

1/4

1/2

1/8

1/3

Correct Answer: A

Solution:

The numbers greater than 6 are 7 and 8. Therefore, the probability is

\frac{2}{8} = \frac{1}{4}

\frac{5}{6}

\frac{7}{8}

\frac{31}{36}

\frac{17}{18}

Correct Answer: A

Solution:

The number of good pens is 144 - 20 = 124. The probability of drawing a good pen is

\frac{124}{144} = \frac{31}{36}

\frac{2}{5}

\frac{3}{10}

\frac{1}{2}

\frac{1}{5}

Correct Answer: B

Solution:

The probability of drawing the first red ball is

\frac{4}{5}

. After drawing one red ball, there are 3 red balls left out of a total of 4 balls. Hence, the probability of drawing another red ball is

\frac{3}{4}

. Therefore, the probability of both balls being red is

\frac{4}{5} \times \frac{3}{4} = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10}

\frac{1}{9}

\frac{1}{12}

\frac{1}{8}

\frac{1}{18}

Correct Answer: B

Solution:

The favorable outcomes for a sum of 9 are (3,6), (4,5), (5,4), and (6,3). Thus, there are 4 favorable outcomes. The total number of outcomes when a die is thrown twice is 36. Therefore, the probability is

\frac{4}{36} = \frac{1}{9}

\frac{1}{4}

\frac{1}{2}

\frac{3}{4}

\frac{1}{3}

Correct Answer: A

Solution:

The possible outcomes when a coin is tossed twice are HH, HT, TH, TT. Only one outcome is two heads (HH), so the probability is

\frac{1}{4}

\frac{3}{8}

\frac{5}{8}

\frac{1}{2}

\frac{1}{4}

Correct Answer: A

Solution:

The probability of drawing a red ball (not black) is the number of red balls divided by the total number of balls. Thus, the probability is

\frac{3}{3+5} = \frac{3}{8}

1/8

1/4

1/2

3/8

Correct Answer: A

Solution:

The possible outcomes for each toss are H or T. Therefore, for three tosses, the total number of outcomes is 2^3 = 8. Only one outcome is HHH. Thus, the probability is 1/8.

1/3

1/2

1/6

1/4

Correct Answer: B

Solution:

The die has 6 faces, and 'A' appears on 2 of them. Therefore, the probability is

\frac{2}{6} = \frac{1}{3}

\frac{99}{100}

\frac{1}{100}

\frac{1}{10}

\frac{1}{2}

Correct Answer: A

Solution:

The probability of not winning is the complement of the probability of winning. Thus, it is

1 - \frac{1}{100} = \frac{99}{100}

0.1389

0.8611

0.5556

0.4444

Correct Answer: B

Solution:

The probability of drawing a non-defective pen is the ratio of non-defective pens to the total number of pens. Thus,

P(\text{not defective}) = \frac{144 - 20}{144} = \frac{124}{144} = 0.8611

4/17

5/17

8/17

9/17

Correct Answer: C

Solution:

The probability of drawing a white marble is given by the ratio of the number of white marbles to the total number of marbles, which is

\frac{8}{17}

1/3

1/2

2/3

5/6

Correct Answer: C

Solution:

The numbers greater than 2 on a die are 3, 4, 5, and 6. Thus, there are 4 favorable outcomes. The probability is

\frac{4}{6} = \frac{2}{3}

1/5

4/5

1/4

1/2

Correct Answer: A

Solution:

There are 5 balls in total, and only 1 of them is blue. Therefore, the probability of drawing a blue ball is 1/5.

1/4

1/3

1/2

3/4

Correct Answer: C

Solution:

The possible outcomes when two coins are tossed are HH, HT, TH, TT. The outcomes with exactly one head are HT and TH. Therefore, the probability is 2/4 = 1/2.

0.25

0.5

0.75

1.0

Correct Answer: B

Solution:

The numbers less than 5 are 1, 2, 3, and 4. So, the probability is

\frac{4}{8} = 0.5

0.38

0.48

0.58

0.42

Correct Answer: A

Solution:

Since the events are complementary, P(Reshma wins) = 1 - P(Sangeeta wins) = 1 - 0.62 = 0.38.

\frac{3}{5}

\frac{1}{2}

\frac{2}{5}

\frac{1}{3}

Correct Answer: C

Solution:

There are 30 students, with 12 being girls. The probability of choosing a girl is

\frac{12}{30} = \frac{2}{5}

1/6

1/3

1/2

1/5

Correct Answer: A

Solution:

Solution:

The numbers less than 3 on a die are 1 and 2. Thus, there are 2 favorable outcomes. The probability is

\frac{2}{6} = \frac{1}{3}

0.2

0.8

0.5

0.1

Correct Answer: A

Solution:

The probability of drawing a blue ball, which is not red, is the number of blue balls divided by the total number of balls. Thus, P(not red) = 1/5 = 0.2.

\frac{9}{17}

\frac{8}{17}

\frac{5}{17}

\frac{12}{17}

Correct Answer: A

Solution:

The total number of marbles is 17. The number of non-white marbles is 5 + 4 = 9. Thus, P(not white) = \frac{9}{17}.

0.25

0.5

0.75

Correct Answer: C

Solution:

The possible outcomes when a coin is tossed twice are HH, HT, TH, TT. The event of getting at least one tail includes HT, TH, and TT. Thus,

P(\text{at least one tail}) = \frac{3}{4} = 0.75

0.1667

0.3333

0.5

0.6667

\frac{1}{3}

Correct Answer: True

Solution:

The sum of the probabilities of all the elementary events of an experiment is 1, as they represent all possible outcomes.

Correct Answer: True

Solution:

The probability of not getting a 5 in one throw is \frac{5}{6}. Thus, the probability of not getting a 5 in two throws is \left(\frac{5}{6}\right)^2 = \frac{25}{36}. Therefore, the probability of getting at least one 5 is 1 - \frac{25}{36} = \frac{11}{36}, which is greater than 0.5.

Correct Answer: True

Solution:

An event that is certain to happen has a probability of 1.

Correct Answer: True

Solution:

A fair die means that each face has an equal chance of landing face up. Therefore, each number from 1 to 6 has a probability of

\frac{1}{6}

Correct Answer: False

Solution:

There are 4 red balls and 1 blue ball, making the probability of drawing a red ball

\frac{4}{5}

and a blue ball

\frac{1}{5}

. These outcomes are not equally likely.

Correct Answer: True

Solution:

A fair die has six faces numbered from 1 to 6, and each face has an equal chance of landing face up. Therefore, the probability of each number appearing is equal.

Correct Answer: True

Solution:

A fair coin has two equally likely outcomes: heads or tails. Therefore, the probability of getting heads is

\frac{1}{2} = 0.5

Correct Answer: True

Solution:

A fair die has numbers 1 to 6, all of which are less than 7, so the probability is 1.

Correct Answer: True

Solution:

The probability of an event is defined as a number

P(E)

such that

0 \leq P(E) \leq 1

Correct Answer: True

Solution:

The die has 6 faces, with 'A' appearing on 2 of them. Therefore, the probability of rolling an 'A' is 2/6, which simplifies to 1/3.

Correct Answer: True

Solution:

When a coin is tossed three times, the total number of outcomes is

2^3 = 8

. The outcome of getting three heads (HHH) is just one of these outcomes, so the probability is

\frac{1}{8}

Correct Answer: False

Solution:

When two coins are tossed, the possible outcomes are HH, TT, and HT (or TH). The probability for each outcome is actually

\frac{1}{4}

for HH,

\frac{1}{4}

for TT, and

Solution:

A sure event is one that is certain to happen, and its probability is always 1.

Correct Answer: False

Solution:

In a fair coin toss, the probability of getting a head is equal to the probability of getting a tail, both being

\frac{1}{2}

Correct Answer: True

Solution:

A standard die has numbers 1 to 6, so there is no possibility of getting a number greater than 6. Therefore, the probability is 0.

Correct Answer: True

Solution:

A die has 6 faces numbered 1 to 6. The odd numbers are 1, 3, and 5, so there are 3 favorable outcomes. The probability is

\frac{3}{6} = \frac{1}{2}

Correct Answer: True

Solution:

A fair die has six faces with numbers 1 to 6. The even numbers are 2, 4, and 6. Thus, the probability of rolling an even number is

\frac{3}{6} = 0.5

Correct Answer: True

Solution:

There are 4 red balls and 1 blue ball, making the probability of drawing a red ball 4/5, which is greater than the probability of drawing a blue ball, 1/5.

Correct Answer: True

Solution:

Probability values range from 0 (impossible event) to 1 (certain event).

Correct Answer: True

Solution:

For any event

E

Solution:

According to the definition of probability, it is a number between 0 and 1, inclusive.

Correct Answer: True

Solution:

A die has 6 faces with numbers 1, 2, 3, 4, 5, and 6. The odd numbers are 1, 3, and 5. Therefore, the probability of getting an odd number is

\frac{3}{6} = \frac{1}{2}

Correct Answer: True

Solution:

When a fair coin is tossed, there are two equally likely outcomes: heads or tails. Thus, the probability of getting a head is

\frac{1}{2} = 0.5

Probability

Learning Objectives

Learning Objectives

Revision Notes & Summary

Probability Notes

Key Concepts

Examples

Important Formulas

Common Mistakes

Tips

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips in Probability

Common Pitfalls

Tips for Success

AI Learning Assistant

Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.In a deck of 52 cards, what is the probability of drawing a spade?

Correct Answer: A

Q2.A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is red?

Correct Answer: A

Q3.A game consists of tossing a fair coin three times. What is the probability of getting exactly two heads?

Correct Answer: B

Q4.A game consists of tossing a coin once. What is the probability of getting a tail?

Correct Answer: B

Q5.A deck of cards is shuffled and one card is drawn at random. What is the probability that it is a face card (Jack, Queen, or King)?

Correct Answer: A

Q6.In a standard deck of 52 cards, what is the probability of drawing a face card?

Correct Answer: A

Q7.A coin is tossed once. What is the probability of getting a tail?

Correct Answer: B

Q8.A fair six-sided die is modified such that the faces show the letters A, B, C, D, E, and A. If the die is thrown once, what is the probability of getting the letter 'A'?

Correct Answer: A

Q9.If two coins are tossed simultaneously, what is the probability of getting at least one head?

Correct Answer: C

Q10.A die is thrown once. What is the probability of getting a number greater than 4?

Correct Answer: B

Q11.A fair die is thrown once. What is the probability of getting a number greater than 4?

Correct Answer: A

Q12.A coin is tossed twice. What is the probability of getting at least one head?

Correct Answer: C

Q13.If a fair coin is tossed 3 times, what is the probability that all tosses result in the same outcome (either all heads or all tails)?

Correct Answer: A

Q14.A game consists of tossing a coin three times. What is the probability that all three tosses result in heads?

Correct Answer: A

Q15.A die is thrown once. What is the probability of getting an even number?

Correct Answer: A

Q16.A spinner is divided into 8 equal sections numbered 1 to 8. What is the probability that it will point at an odd number?

Correct Answer: B

Q17.A spinner is divided into 8 equal sections numbered 1 to 8. What is the probability that it will land on a number greater than 6?

Correct Answer: A

Q18.If a bag contains 4 red balls, 3 blue balls, and 2 green balls, what is the probability of drawing a blue ball?

Correct Answer: C

Q19.In a standard deck of 52 playing cards, what is the probability of drawing a face card?

Correct Answer: A

Q20.A game consists of tossing a one rupee coin 3 times. What is the probability of getting exactly two heads?

Correct Answer: B

Q21.A die is thrown once. What is the probability of getting a number less than 4?

Correct Answer: B

Q22.A fair die is thrown twice. What is the probability that the sum of the numbers on the two dice will be 7?

Correct Answer: A

Q23.A die is thrown once. What is the probability of getting a number less than 3?

Correct Answer: A

Q24.A die is designed such that it has faces labeled with the letters A, B, C, D, E, and A. If the die is thrown once, what is the probability of getting the letter 'A'?

Correct Answer: B

Q25.In a game, a spinner is divided into 8 equal sections numbered 1 to 8. What is the probability that the spinner lands on a prime number?

Correct Answer: B

Q26.A fair die is thrown twice. What is the probability that the sum of the numbers on the two dice is 9?

Correct Answer: B

Q27.In a game, a spinner is divided into 8 equal sections numbered 1 to 8. What is the probability that the spinner will land on a number that is a multiple of 4?

Correct Answer: B

Q28.A box contains 4 red, 6 blue, and 5 yellow marbles. If two marbles are drawn at random, what is the probability that both marbles are blue?

Correct Answer: A

Q29.A game consists of spinning a wheel with 8 equal sections numbered 1 to 8. What is the probability that it will land on a number greater than 6?

Correct Answer: A

Q30.A lot consists of 144 ball pens of which 20 are defective and the others are good. If a pen is drawn at random, what is the probability that it is good?

Correct Answer: A

Q31.A bag contains 4 red balls and 1 blue ball. If two balls are drawn at random without replacement, what is the probability that both balls are red?

Correct Answer: B

Q32.A die is thrown twice. What is the probability that the sum of the numbers obtained is 9?