Probability

Probability is the mathematical measure of the likelihood of an event occurring. It quantifies uncertainty.

• Experiment: An action or process that results in well-defined outcomes.
• Random Experiment: An experiment where all possible outcomes are known, but the exact outcome cannot be predicted in advance.
• Examples: Tossing a coin, rolling a die, drawing a card from a deck.
• Outcome: A single possible result of an experiment.
• Example: Getting a 'Head' when tossing a coin, getting a '3' when rolling a die.
• Sample Space (S): The set of all possible outcomes of a random experiment.
• Examples:
• Tossing a coin: \(S = \{H, T\}\)
• Rolling a die: \(S = \{1, 2, 3, 4, 5, 6\}\)
• Tossing two coins: \(S = \{HH, HT, TH, TT\}\)
• Event (E): A subset of the sample space. It is a collection of one or more outcomes.
• Examples:
• Getting an even number when rolling a die: \(E = \{2, 4, 6\}\)
• Getting at least one Head when tossing two coins: \(E = \{HH, HT, TH\}\)
• Elementary Event: An event having only one outcome of the random experiment.
• Example: Getting a '6' when rolling a die is an elementary event.
• Compound Event: An event having more than one outcome of the random experiment.
• Example: Getting an even number when rolling a die is a compound event.

The classical definition of probability is applicable when all outcomes of an experiment are equally likely.

• Equally Likely Outcomes: Outcomes are equally likely if each has the same chance of occurring.
• Example: In a fair coin toss, getting a Head or a Tail are equally likely.

• Formula for Probability of an Event E:

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

• Steps to calculate probability:

1. Identify the random experiment.
2. Determine the sample space \(S\) and find \(n(S)\).
3. Identify the event \(E\) and find \(n(E)\) (number of outcomes favorable to E).
4. Apply the formula \(P(E) = \frac{n(E)}{n(S)}\).

• Example: What is the probability of rolling an even number on a fair six-sided die?
• Experiment: Rolling a die.
• Sample Space \(S = \{1, 2, 3, 4, 5, 6\}\), so \(n(S) = 6\).
• Event E: Getting an even number \(E = \{2, 4, 6\}\), so \(n(E) = 3\).
• \(P(E) = \frac{3}{6} = \frac{1}{2}\).

The probability of any event \(E\) is a number between 0 and 1, inclusive.

• Probability Scale:
• 0 (Impossible Event): An event that cannot occur. \(P(E) = 0\).
• Example: Getting a '7' when rolling a single six-sided die. \(n(E) = 0\), \(P(E) = 0/6 = 0\).
• 1 (Sure/Certain Event): An event that is certain to occur. \(P(E) = 1\).
• Example: Getting a number less than 7 when rolling a single six-sided die. \(E = \{1, 2, 3, 4, 5, 6\}\), \(n(E) = 6\), \(P(E) = 6/6 = 1\).
• 0.5 (Equally Likely): When the chances of an event occurring or not occurring are equal.
• Example: Getting a Head in a coin toss. \(P(H) = 1/2 = 0.5\).

• Types of Events:
• Impossible Event: As described above, \(P(E) = 0\).
• Sure Event (Certain Event): As described above, \(P(E) = 1\).
• Simple Event: An event with only one outcome.
• Compound Event: An event with more than one outcome.
• Complementary Events: Discussed in detail in Topic 4.
• Mutually Exclusive Events: Events that cannot occur at the same time. If \(E_1\) and \(E_2\) are mutually exclusive, then \(P(E_1 \text{ and } E_2) = 0\).
• Example: When rolling a die, getting an even number and getting an odd number are mutually exclusive.
• Exhaustive Events: A set of events is exhaustive if at least one of them must occur. The union of all exhaustive events forms the sample space.
• Example: When rolling a die, getting an even number or getting an odd number are exhaustive events. \(E_{even} = \{2,4,6\}\), \(E_{odd} = \{1,3,5\}\). \(E_{even} \cup E_{odd} = S\).

The complement of an event E, denoted as \(E'\) or \(\bar{E}\) or \(E^c\), is the event that E does not occur.

• Relationship: The event \(E\) and its complement \(E'\) together make up the entire sample space.
• \(E \cup E' = S\)
• \(E \cap E' = \emptyset\) (They are mutually exclusive)

• Formula for Complementary Events:

\(P(E') = 1 - P(E)\) or equivalently, \(P(E) + P(E') = 1\)

• Derivation: Since \(E\) and \(E'\) are mutually exclusive and exhaustive, the sum of their probabilities must be 1.
• \(n(E) + n(E') = n(S)\)
• Dividing by \(n(S)\) on both sides:

\(\frac{n(E)}{n(S)} + \frac{n(E')}{n(S)} = \frac{n(S)}{n(S)}\)

• \(P(E) + P(E') = 1\)

• Example: The probability of rain on a particular day is 0.7. What is the probability that it will not rain?
• Let E be the event that it rains. \(P(E) = 0.7\).
• Let E' be the event that it does not rain.
• \(P(E') = 1 - P(E) = 1 - 0.7 = 0.3\).

• Usefulness: This formula is particularly useful when it's easier to calculate the probability of an event not happening than the event itself (e.g., 'at least one' type problems).

Many probability problems involve standard objects like coins, dice, and playing cards. Understanding their sample spaces is crucial.

Coins

• One Coin: \(S = \{H, T\}\), \(n(S) = 2\)
• Two Coins: \(S = \{HH, HT, TH, TT\}\), \(n(S) = 4\)
• Three Coins: \(S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}\), \(n(S) = 8\)
• Generalization: For \(n\) coins, \(n(S) = 2^n\).

Dice

• One Die: \(S = \{1, 2, 3, 4, 5, 6\}\), \(n(S) = 6\)
• Two Dice: Each outcome is an ordered pair \((d_1, d_2)\) where \(d_1, d_2 \in \{1, ..., 6\}\).
• \(n(S) = 6 \times 6 = 36\).
• Example outcomes: (1,1), (1,2), ..., (6,6).
• Common events: Sum of numbers, difference, product, specific numbers on each die.

Playing Cards

• A standard deck has 52 cards.
• Suits: 4 suits – Spades (♠), Clubs (♣), Hearts (♥), Diamonds (♦).
• Spades and Clubs are black cards (13 each, total 26).
• Hearts and Diamonds are red cards (13 each, total 26).
• Ranks: 13 ranks in each suit – A (Ace), 2, 3, 4, 5, 6, 7, 8, 9, 10, J (Jack), Q (Queen), K (King).
• Face Cards: Jack, Queen, King are face cards.
• Total face cards: \(3 \text{ ranks} \times 4 \text{ suits} = 12\) face cards.
• Ace: Can be considered high or low depending on the game, but usually counted as a distinct rank.

• Key Card Probabilities:
• \(P(\text{Red Card}) = 26/52 = 1/2\)
• \(P(\text{Black Card}) = 26/52 = 1/2\)
• \(P(\text{King}) = 4/52 = 1/13\)
• \(P(\text{Face Card}) = 12/52 = 3/13\)
• \(P(\text{Ace of Spades}) = 1/52\)

Probability is not just a theoretical concept; it has wide-ranging real-world applications.

• Weather Forecasting: Meteorologists use probability to predict the chance of rain, snow, or sunshine. (e.g., "There's a 70% chance of rain today.")
• Sports: Coaches and analysts use probability to assess team performance, predict game outcomes, and make strategic decisions. (e.g., probability of winning a match, a player scoring a goal).
• Insurance: Insurance companies use probability to calculate premiums based on the likelihood of an event (e.g., car accident, house fire, health issues) occurring.
• Medical Diagnosis: Doctors use probability to determine the likelihood of a patient having a particular disease based on symptoms and test results.
• Genetics: Probability is fundamental in understanding inheritance patterns and predicting the likelihood of offspring inheriting certain traits or genetic disorders.
• Quality Control: Manufacturers use probability to test a sample of products to ensure the quality of the entire batch.
• Finance and Stock Market: Investors use probability to assess risks and potential returns of investments.
• Gaming and Gambling: The entire industry is built on probability, from card games to lotteries, where odds are calculated to determine payouts.