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Probability

Action PlanNotesQuestionsMock-Test
Study Action Plan (Sorted by Difficulty)
Elementary Events
Accuracy: 53%
Total Attempted92
Total Correct49
Total Incorrect43
Difficulty Level:
Medium
Complementary Events
Accuracy: 61%
Total Attempted46
Total Correct28
Total Incorrect18
Difficulty Level:
Medium
Probability of Sure and Impossible Events
Accuracy: 62%
Total Attempted61
Total Correct38
Total Incorrect23
Difficulty Level:
Medium
Empirical vs Theoretical Probability
Accuracy: 67%
Total Attempted9
Total Correct6
Total Incorrect3
Difficulty Level:
Medium
Probability from Real Scenarios (Cards, Dice, Multiple Trials)
Accuracy: 69%
Total Attempted124
Total Correct85
Total Incorrect39
Difficulty Level:
Medium
Sample Space & Outcome Enumeration
Accuracy: 73%
Total Attempted83
Total Correct61
Total Incorrect22
Difficulty Level:
Medium
Classical Probability
Accuracy: 79%
Total Attempted138
Total Correct109
Total Incorrect29
Difficulty Level:
Low

Learning Objectives

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Learning Objectives

  • Understand the concept of Classical Probability and calculate the probability of an event using the formula: P(E)=Number of outcomes favorable to ENumber of all possible outcomesP(E) = \frac{\text{Number of outcomes favorable to } E}{\text{Number of all possible outcomes}}, assuming equally likely outcomes.
  • Apply the principle of Complementary Events where for any event EE, the relationship P(E)+P(not E)=1P(E) + P(\text{not } E) = 1 holds true.
  • Identify Elementary Events as events with only one outcome and recognize that the sum of the probabilities of all elementary events in an experiment is 1.
  • Distinguish between Probability of Sure and Impossible Events, where the probability of a sure event is 1 and that of an impossible event is 0.
  • Differentiate between Empirical and Theoretical Probability, understanding that empirical probability is based on actual outcomes, while theoretical probability is based on assumptions and predictions.

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Revision Notes & Summary

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Chapter Notes

Classical Probability

  • Definition: The theoretical probability of an event EE is defined as the ratio of the number of outcomes favorable to EE to the number of all possible outcomes, assuming equally likely outcomes.
  • Formula: P(E)=Number of outcomes favorable to ENumber of all possible outcomesP(E) = \frac{\text{Number of outcomes favorable to } E}{\text{Number of all possible outcomes}}

Complementary Events

  • Definition: For any event EE, the probability of EE plus the probability of not EE equals 1.
  • Formula: P(E)+P(not E)=1P(E) + P(\text{not } E) = 1
  • Complementary Events: EE and not EE are complementary events.

Elementary Events

  • Definition: An event with only one outcome is called an elementary event.
  • Property: The sum of the probabilities of all elementary events of an experiment is 1.

Probability of Sure and Impossible Events

  • Sure Event: The probability of a sure event is 1.
  • Impossible Event: The probability of an impossible event is 0.

Empirical vs Theoretical Probability

  • Empirical Probability: Based on actual outcomes of experiments.
  • Theoretical Probability: Based on assumptions and predictions.
  • Convergence: As the number of trials increases, empirical probability tends to converge to theoretical probability.

Key Properties

  • Range: For any event EE, 0≤P(E)≤10 \leq P(E) \leq 1.
  • Elementary Event Sum: The sum of probabilities of all elementary events is 1.

Examples

  • Coin Toss: Probability of getting a head when a coin is tossed is P(Head)=12P(\text{Head}) = \frac{1}{2}.
  • Dice Roll: Probability of getting a number greater than 4 when a die is rolled is P(>4)=26=13P(>4) = \frac{2}{6} = \frac{1}{3}.
  • Card Draw: Probability of drawing an ace from a deck of cards is P(Ace)=452=113P(\text{Ace}) = \frac{4}{52} = \frac{1}{13}.

Important Notes

  • Empirical Probability: Requires a large number of trials and may not be feasible in all situations.
  • Theoretical Probability: Useful when assumptions of equally likely outcomes are valid.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

313\frac{3}{13}

B.

14\frac{1}{4}

C.

113\frac{1}{13}

D.

12\frac{1}{2}
Correct Answer: A

Solution:

There are 12 face cards (3 face cards per suit) in a deck of 52 cards. The probability of drawing a face card is 1252=313\frac{12}{52} = \frac{3}{13}.

Chapter Concept:

Elementary Events

A.

452\frac{4}{52}

B.

4852\frac{48}{52}

C.

113\frac{1}{13}

D.

1213\frac{12}{13}
Correct Answer: D

Solution:

There are 4 aces in a deck of 52 cards. The probability of drawing an ace is 452=113\frac{4}{52} = \frac{1}{13}. Therefore, the probability of not drawing an ace is 1−113=12131 - \frac{1}{13} = \frac{12}{13}.

Chapter Concept:

Complementary Events

A.

0.375

B.

0.4

C.

0.5

D.

0.625
Correct Answer: A

Solution:

The probability of selecting a girl is 2540=0.625\frac{25}{40} = 0.625. Therefore, the probability of not selecting a girl is 1−0.625=0.3751 - 0.625 = 0.375.

Chapter Concept:

Complementary Events

A.

16\frac{1}{6}

B.

13\frac{1}{3}

C.

12\frac{1}{2}

D.

23\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, so the probability is 26=13\frac{2}{6} = \frac{1}{3}.

Chapter Concept:

Classical Probability

A.

326\frac{3}{26}

B.

113\frac{1}{13}

C.

352\frac{3}{52}

D.

14\frac{1}{4}
Correct Answer: A

Solution:

There are 6 red face cards (3 hearts and 3 diamonds) in a deck of 52 cards. The probability is 652=326\frac{6}{52} = \frac{3}{26}.

Chapter Concept:

Classical Probability

A.

19\frac{1}{9}

B.

536\frac{5}{36}

C.

16\frac{1}{6}

D.

112\frac{1}{12}
Correct Answer: B

Solution:

The possible outcomes for a sum of 9 are (3,6), (4,5), (5,4), (6,3). There are 4 favorable outcomes out of 36 possible outcomes. Thus, the probability is 436=19\frac{4}{36} = \frac{1}{9}.

Chapter Concept:

Probability from Real Scenarios (Cards, Dice, Multiple Trials)

A.

15\frac{1}{5}

B.

45\frac{4}{5}

C.

14\frac{1}{4}

D.

35\frac{3}{5}
Correct Answer: A

Solution:

There is only 1 blue ball in a total of 5 balls. Thus, the probability of drawing a blue ball is 15\frac{1}{5}.

Chapter Concept:

Elementary Events

A.

16\frac{1}{6}

B.

13\frac{1}{3}

C.

56\frac{5}{6}

D.

12\frac{1}{2}
Correct Answer: A

Solution:

The possible outcomes when a die is rolled are 1, 2, 3, 4, 5, and 6. The number of outcomes greater than 5 is 1 (only 6). Therefore, the probability is 16\frac{1}{6}.

Chapter Concept:

Classical Probability

A.

0.88

B.

0.92

C.

0.96

D.

0.98
Correct Answer: C

Solution:

Sujatha accepts shirts with minor or no defects. Thus, she accepts 88 + 8 = 96 shirts. The probability is 96100=0.96\frac{96}{100} = 0.96.

Chapter Concept:

Sample Space & Outcome Enumeration

A.

0.0769

B.

0.1538

C.

0.8462

D.

0.9231
Correct Answer: C

Solution:

There are 4 aces in a deck of 52 cards, so the probability of drawing an ace is 452=0.0769\frac{4}{52} = 0.0769. Therefore, the probability of not drawing an ace is 1−0.0769=0.92311 - 0.0769 = 0.9231.

Chapter Concept:

Complementary Events

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Probability

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True or False

Correct Answer: False

Solution:

An elementary event is defined as an event with only one outcome.

Chapter Concept :

Elementary Events

Correct Answer: True

Solution:

The bag contains 4 red balls and 1 blue ball, making a total of 5 balls. The probability of drawing a red ball is 45\frac{4}{5} since there are 4 favorable outcomes.

Chapter Concept :

Probability from Real Scenarios (Cards, Dice, Multiple Trials)

Correct Answer: False

Solution:

The probability of getting a number greater than 6 on a standard die is 0, as the highest number on a die is 6.

Chapter Concept :

Classical Probability

Correct Answer: False

Solution:

The probability of an impossible event is 0, as there are no outcomes favorable to such an event.

Chapter Concept :

Classical Probability

Correct Answer: False

Solution:

The probability of a sure event is 1, not 0.

Chapter Concept :

Probability of Sure and Impossible Events

Correct Answer: True

Solution:

The probability of an event EE and its complement E‾\overline{E} must sum to 1, as they encompass all possible outcomes of the event.

Chapter Concept :

Complementary Events

Correct Answer: True

Solution:

When a die is thrown, the numbers greater than 4 are 5 and 6. Thus, there are 2 favorable outcomes out of 6 possible outcomes, giving a probability of 2/6 = 1/3.

Chapter Concept :

Classical Probability

Correct Answer: False

Solution:

An event with only one outcome is called an elementary event.

Chapter Concept :

Elementary Events

Correct Answer: False

Solution:

The probability of an impossible event is 0, not 1. An impossible event is one that cannot occur under any circumstances.

Chapter Concept :

Probability of Sure and Impossible Events

Correct Answer: True

Solution:

When a fair die is tossed, the possible outcomes are 1, 2, 3, 4, 5, and 6, each of which is equally likely.

Chapter Concept :

Sample Space & Outcome Enumeration

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