Probability E-Note for senior secondary school mathematics by Edwin Ogie Library 3

Understanding Probability: Definition, Types, and Solved Examples

Objectives

By the end of this lesson, you should be able to:

1. Define and explain key terms used in probability.
2. Provide practical examples of each term.
3. Solve simple problems on mutually exclusive, independent, and complementary events.
4. Solve probability problems involving experiments with or without replacement.

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What is Probability?

Probability is a mathematical concept that measures the likelihood of an event occurring. It is expressed as a ratio of the number of favorable outcomes to the total number of possible outcomes.

Mathematical Representation

$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

The probability of an event always lies between 0 and 1:

• 0 means the event is impossible.
• 1 means the event is certain to happen.
• Any value between 0 and 1 represents the likelihood of occurrence.

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Examples and Solutions

Example 1: Rolling a Die

Find the probability of getting:
a) A 1
b) A 2
c) A 3
in a single toss of a fair die.

Solution:

A fair die has six sides numbered 1 to 6. Each number appears once, making the probability of rolling any specific number:

$$P(1) = \frac{1}{6}, \quad P(2) = \frac{1}{6}, \quad P(3) = \frac{1}{6}$$

Since all outcomes are equally likely, the probability of rolling any number is $\frac{1}{6}$.

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Example 2: Tossing a Coin

What is the probability of getting a head when tossing a fair coin?

Solution:

A fair coin has two possible outcomes: Head (H) and Tail (T).

$$P(H) = \frac{1}{2}$$

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Example 3: Probability of Factors of 6

What is the probability of getting a factor of 6 when rolling a fair die?

Solution:

A fair die has numbers: 1, 2, 3, 4, 5, 6.
The factors of 6 are: 1, 2, 3, and 6 (four numbers).

$$P(\text{factor of 6}) = \frac{4}{6} = \frac{2}{3}$$

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Example 4: Colored Balls in a Basket

A basket contains 4 white, 3 red, and 5 blue balls. If a ball is picked at random, find the probability that it is:
a) Red
b) Green
c) White
d) Not Red

Solution:

Total number of balls = $4 + 3 + 5 = 12$

$$P(Red) = \frac{3}{12} = \frac{1}{4}, \quad P(Green) = 0, \quad P(White) = \frac{4}{12} = \frac{1}{3}$$

For "Not Red":

$$P(\text{Not Red}) = P(White) + P(Blue) = \frac{1}{3} + \frac{5}{12} = \frac{7}{12}$$

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Probability of an Event NOT Happening

The probability of an event not happening is calculated as:

$$P(\text{not X}) = 1 - P(X)$$

Example 5: Not Getting a Certain Color

A bag contains 10 balls:

• 5 Black
• 3 Yellow
• The rest are Green

A ball is chosen at random. Find the probability that the ball is NOT:
a) Black
b) Yellow
c) Green

Solution:

Total balls = 10
Green balls = $10 - (5 + 3) = 2$

$$P(\text{Not Black}) = 1 - P(\text{Black}) = 1 - \frac{5}{10} = \frac{1}{2}$$ $$P(\text{Not Yellow}) = 1 - P(\text{Yellow}) = 1 - \frac{3}{10} = \frac{7}{10}$$ $$P(\text{Not Green}) = 1 - P(\text{Green}) = 1 - \frac{2}{10} = \frac{4}{5}$$

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Types of Probability

1. Experimental Probability

• Based on actual experiments or trials.
• More reliable because results are derived from real-life observations.
• Example: Tossing a coin or rolling a die multiple times and recording the outcomes.

2. Theoretical Probability

• Based on mathematical calculations rather than experiments.
• Less reliable because it assumes perfect conditions.
• Example: Predicting a student’s test score or the future price of an item.

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Example 6: Probability from a Frequency Table

The results of rolling a fair die 150 times are summarized below:

Die Outcome	Frequency
1	12
2	18
3	x
4	30
5	2x
6	45
Total	150

Find the probability of rolling a 5.

Solution:

First, determine $x$.

$$12 + 18 + x + 30 + 2x + 45 = 150$$ $$105 + 3x = 150$$ $$3x = 45, \quad x = 15$$

Now, find $P(5)$:

$$P(5) = \frac{2x}{150} = \frac{2(15)}{150} = \frac{30}{150} = \frac{1}{5}$$

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Conclusion

Probability helps us measure uncertainty in various real-life situations, from games of chance to predicting future events. By practicing more problems and understanding the rules, you can master probability calculations effectively.

Would you like to try more examples? Drop your questions in the comment section!

Probability Quiz (CBT JAMB)

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