Probability of an Event

Definition: A probability set is the set of all outcomes/results from the experiment being performed. For example, when tossing once a fair coin, the expected outcomes are either head (H) or tail (T). Thus the probability set is:

$$S = \{H, T\}$$

Also, if a fair die is tossed once, the expected outcome is one number among 1, 2, 3, 4, 5, 6. Thus:

$$S = \{1, 2, 3, 4, 5, 6\}$$

An event is a specified outcome from the probability set. For example, getting a head (H) when tossing a coin is an event and a subset of the probability set. Thus:

$$S = \{H, T\} \quad \text{and} \quad E = \{H\}$$

An event may or may not occur. For instance, if the event is getting a head but a tail occurs, then the event did not occur. It is denoted by $E'$ (the complement of E). Thus:

$$E' = \{T\}$$

NB: A probability set is also called a sample space.

A die is tossed once and the results are recorded. Find:

1. The probability set (sample space).
2. The event that an even number occurs.
3. The event that an even number does not occur.

Solution: Sample space:

$$S = \{1, 2, 3, 4, 5, 6\}$$

The event that an even number occurs:

$$E = \{2, 4, 6\}$$

The event that an even number does not occur:

$$E' = \{1, 3, 5\}$$

Give the probability set for selecting even numbers less than 20. Solution:

$$S = \{2, 4, 6, 8, 10, 12, 14, 16, 18\}$$

Give the probability set for not selecting an even number from a set of counting numbers less than 9. Solution: Sample space:

$$S = \{1, 2, 3, 4, 5, 6, 7, 8\}$$

Event of selecting even number:

$$E = \{2, 4, 6, 8\}$$

Event of not selecting an even number:

$$E' = \{1, 3, 5, 7\}$$

Example: Tossing a fair coin once gives the outcomes:

$$S = \{H, T\}$$

Example: Tossing a fair die once gives:

$$S = \{1, 2, 3, 4, 5, 6\}$$

Definition: The probability of an event is the ratio between the number of times the event has occurred to the total number of experiments performed.

$$P(E) = \frac{\text{Number of times } E \text{ has occurred}}{\text{Total number of times the experiment has been done}}$$ $$\text{Or} \quad P(E) = \frac{n(E)}{n(S)}$$

where $P(E)$ is the probability of an event $E$, $n(E)$ is the number of elements in $E$ which is actually the number of times $E$ has occurred and $n(S)$ is the number of elements in the sample space which is the total number of possible outcomes of the experiment.

$$\text{NB:} \quad \frac{n(E)}{n(S)} \quad \text{is the ratio called the observed relative frequency of the event which is the probability of an event } E \text{ to occur}.$$

The probability found by experimenting is referred to as experimental probability.

A drawing pin was tossed 1000 times. The number of tosses where the pin fell flat was 563. Find the probability that it falls flat. Solution: The probability $P(E)$ is given by:

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

Thus,

$$P(E) = \frac{563}{1000} = 0.563$$

5% of torch bulbs manufactured by a certain factory were defective. What is the probability that when a bulb is tested, it will be defective? Solution: Given that 5% of bulbs are defective:

$$P(E) = 5\% = \frac{5}{100} = 0.05$$

A piece of chalk is picked from a box containing 5 identical pieces, 2 of which are red and 3 are white. Find the probability that the piece picked is red. Solution: Number of red pieces = 2 Total pieces = 5 Therefore,

$$P(\text{Red}) = \frac{2}{5} = 0.4$$

Find the probability that a king appears when drawing a single card from an ordinary deck of 52 cards. Solution: There are 4 kings in a deck of 52 cards. Thus,

$$P(\text{King}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769$$

What is the probability of not getting an even number when a fair die is tossed? Solution: Sample space:

$$S = \{1, 2, 3, 4, 5, 6\}$$

Even numbers are:

$$\{2, 4, 6\}$$

Thus, the probability of getting an even number:

$$P(\text{Even}) = \frac{3}{6} = \frac{1}{2}$$

Hence, the probability of not getting an even number is:

$$P(\text{Not Even}) = 1 - P(\text{Even}) = 1 - \frac{1}{2} = \frac{1}{2} = 0.5$$