Probability

14.1 Event

This section introduces the fundamental concept of an event in probability theory, building upon the earlier notions of random experiments and sample spaces. A random experiment is an action or process that leads to one of several possible outcomes, which are collectively represented by the sample space S. Each outcome is a sample point in S. An event is defined as any subset of the sample space, representing a collection of outcomes of interest. For example, consider the experiment of tossing a coin twice. The sample space is S = {HH, HT, TH, TT}, where H and T denote heads and tails respectively. An event E could be 'exactly one head appears', which corresponds to the subset E = {HT, TH} of S. This shows that events are subsets of the sample space and can range from single outcomes to multiple outcomes. The section further illustrates this with a table showing various event descriptions and their corresponding subsets of S, such as the event 'number of tails is exactly 2' corresponding to {TT}, or 'number of tails is more than two' corresponding to the empty set φ, which is an impossible event. This conceptualization of events as subsets of sample space is foundational for defining probabilities and analyzing random experiments. **Table on page 1 (7×2)** | Description of events | Corresponding subset of ‘S’ | | --- | --- | | Number of tails is exactly 2 | A = {TT} | | Number of tails is atleast one | B = {HT, TH, TT} | | Number of heads is atmost one | C = {HT, TH, TT} | | Second toss is not head | D = { HT, TT} | | Number of tails is atmost two | S = {HH, HT, TH, TT} | | Number of tails is more than two | φ | **Table on page 18 (6×8)** | Assignment | ω_{1} | ω_{2} | ω_{3} | ω_{4} | ω_{5} | ω_{6} | ω_{7} | | --- | --- | --- | --- | --- | --- | --- | --- | | (a) | 0.1 | 0.01 | 0.05 | 0.03 | 0.01 | 0.2 | 0.6 | | (b) | 1/7 | 1/7 | 1/7 | 1/7 | 1/7 | 1/7 | 1/7 | | (c) | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | | (d) | -0.1 | 0.2 | 0.3 | 0.4 | -0.2 | 0.1 | 0.3 | | (e) | 1/14 | 2/14 | 3/14 | 4/14 | 5/14 | 6/14 | 15/14 | **Table on page 19 (4×5)** | | P(A) | P(B) | P(A∩B) | P(A∪B) | | --- | --- | --- | --- | --- | | (i) | 1/3 | 1/5 | 1/15 | ... | | (ii) | 0.35 | ... | 0.25 | 0.6 | | (iii) | 0.5 | 0.35 | ... | 0.7 | **Table on page 23 (6×4)** | S. No. | Name | Sex | Age in years | | --- | --- | --- | --- | | 1. | Harish | M | 30 | | 2. | Rohan | M | 33 | | 3. | Sheetal | F | 46 | | 4. | Alis | F | 28 | | 5. | Salim | M | 41 |

• A random experiment has a sample space S consisting of all possible outcomes.
• An event is any subset E of the sample space S.
• Events can represent single or multiple outcomes.
• Example: Tossing two coins, event 'exactly one head' corresponds to E = {HT, TH}.
• The empty set φ represents an impossible event.
• The whole sample space S represents a sure event.
• 📌 Random experiment: An action or process with uncertain outcomes.
• 📌 Sample space (S): The set of all possible outcomes of a random experiment.
• 📌 Event: Any subset of the sample space representing outcomes of interest.

14.1.1 Occurrence of an event

This subsection explains the condition under which an event is said to have occurred in the context of a random experiment. Consider the experiment of throwing a die with sample space S = {1, 2, 3, 4, 5, 6}. Let event E be 'a number less than 4 appears', so E = {1, 2, 3}. If the actual outcome ω of the experiment is such that ω ∈ E, for example, if the die shows 1, 2, or 3, then the event E is said to have occurred. Conversely, if the outcome ω ∉ E, say the die shows 4, 5, or 6, then the event E has not occurred. Thus, the occurrence of an event depends on whether the outcome of the experiment belongs to the subset defining the event. This clear criterion helps in identifying event occurrence in practical situations and forms the basis for assigning probabilities to events.

• An event E occurs if the outcome ω of the experiment is in E.
• If ω ∉ E, the event E does not occur.
• Example: Throwing a die, event E = 'number less than 4' corresponds to E = {1, 2, 3}.
• If outcome is 1, 2, or 3, event E occurs.
• If outcome is 4, 5, or 6, event E does not occur.
• 📌 Outcome (ω): The actual result of a random experiment.
• 📌 Occurrence of an event: When the outcome belongs to the event subset.