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Probability - Events and definition of probability

Grade 10ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Random Experiment: An operation or action which can produce some well-defined outcomes, but the exact result cannot be predicted with certainty. Visualizing a fair coin toss, we see two possible outcomes: 'Heads' or 'Tails', but cannot pre-determine the landing face.

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Sample Space (SS): The set of all possible outcomes of a random experiment. For a single die roll, the sample space is S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}. Visualize this as a set of all possible branches in a probability tree diagram.

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Event (EE): A subset of the sample space representing one or more specific outcomes. If we want an 'even number' from a die roll, E={2,4,6}E = \{2, 4, 6\}. In a Venn diagram, the event EE is represented as a closed circle or loop entirely contained within the rectangle of the sample space SS.

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Equally Likely Outcomes: Outcomes are said to be equally likely if none of them has a higher preference or chance of occurring than others. Imagine a fair spinner divided into 8 identical sectors; each sector has the same area, making the chance of landing on any one color equally likely.

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Probability Scale: The probability P(E)P(E) of any event EE is a number such that 0≤P(E)≤10 \le P(E) \le 1. On a horizontal number line, 00 represents an 'Impossible Event' (e.g., drawing a blue card from a deck of only red cards) and 11 represents a 'Sure Event' (e.g., the sun rising in the east).

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Complementary Events: For every event EE, there is an event 'not EE', denoted by E′E' or E‾\overline{E}. Visualized as the entire region inside the sample space rectangle that is outside the circle of event EE. The sum of the probability of an event and its complement is always 11.

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Standard Deck of Cards: A standard deck contains 5252 cards divided into 44 suits: Hearts (red), Diamonds (red), Spades (black), and Clubs (black). Each suit has 1313 cards, including 'Face Cards' (Jack, Queen, King). Visualizing the grid of 4 rows (suits) and 13 columns (ranks) helps in identifying subsets like 'all red cards' or 'all kings'.

📐Formulae

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

P(E)=n(E)n(S)P(E) = \frac{n(E)}{n(S)}

0≤P(E)≤10 \le P(E) \le 1

P(E)+P(E′)=1P(E) + P(E') = 1

P(E′)=1−P(E)P(E') = 1 - P(E)

P(Sure Event)=1P(\text{Sure Event}) = 1

P(Impossible Event)=0P(\text{Impossible Event}) = 0

💡Examples

Problem 1:

A bag contains 5 red balls, 8 white balls, and 4 green balls. One ball is drawn at random. Find the probability that the ball drawn is (i) red (ii) not green.

Solution:

Step 1: Find the total number of outcomes. Total balls n(S)=5+8+4=17n(S) = 5 + 8 + 4 = 17.

Step 2: Solve for (i) Red ball. Number of red balls n(R)=5n(R) = 5. Probability P(R)=n(R)n(S)=517P(R) = \frac{n(R)}{n(S)} = \frac{5}{17}.

Step 3: Solve for (ii) Not green ball. Number of green balls n(G)=4n(G) = 4. Probability of green P(G)=417P(G) = \frac{4}{17}. Probability of 'not green' P(G′)=1−P(G)=1−417=1317P(G') = 1 - P(G) = 1 - \frac{4}{17} = \frac{13}{17}.

Explanation:

We first calculate the total possible outcomes by summing all items. We then use the basic probability definition for the first part and the complementary event formula for the second part.

Problem 2:

A card is drawn from a well-shuffled deck of 52 cards. Find the probability that the card drawn is (i) a face card (ii) a red king.

Solution:

Step 1: Total number of outcomes n(S)=52n(S) = 52.

Step 2: Solve for (i) Face card. In each suit, there are 3 face cards (Jack, Queen, King). Since there are 4 suits, total face cards n(F)=3×4=12n(F) = 3 \times 4 = 12. Probability P(F)=1252=313P(F) = \frac{12}{52} = \frac{3}{13}.

Step 3: Solve for (ii) Red king. There are 2 red suits (Hearts and Diamonds), and each has 1 king. Total red kings n(K)=2n(K) = 2. Probability P(K)=252=126P(K) = \frac{2}{52} = \frac{1}{26}.

Explanation:

This solution relies on understanding the structure of a standard 52-card deck, specifically the count of face cards across all suits and the distribution of colors among the kings.