Probability is a fundamental concept in mathematics, used to measure the likelihood of events occurring. It plays a crucial role in various fields, including statistics, finance, and science, helping us make informed decisions based on uncertain outcomes.

Probability Scale

• 0 (Impossible): Event cannot happen.
• 1 (Certain): Event will definitely happen.
• Values between 0 and 1 represent varying levels of likelihood.

Probability Notation

• P(A): Probability of event A.

Worked Examples

Example 1: Flipping a Coin

Consider rolling a standard six-sided dice.

• Success event: Rolling a four.
• Total possible outcomes: 6 (one for each side of the dice).

Calculation:

$P(\text{Four}) = \frac{\text{Number of ways to achieve a four}}{\text{Total possible outcomes}} = \frac{1}{6}$

This shows that the probability of rolling a four is approximately 0.167, indicating it is less likely to occur compared to flipping heads on a coin.

Image courtesy of Pixabay

Example 2: Rolling a Dice

Consider rolling a standard six-sided dice.

• Success event: Rolling a four.
• Total possible outcomes: 6 (one for each side of the dice).

Calculation:

$ P(\text{Four}) = \frac{\text{Number of ways to achieve a four}}{\text{Total possible outcomes}} = \frac{1}{6}$<p></p><p>This shows that the probability of rolling a four is approximately 0.167, indicating it is less likely to occur compared to flipping heads on a coin.</p><img src="https://tutorchase-production.s3.eu-west-2.amazonaws.com/2658ce21-2049-42ff-b388-ee917fa999de-file.png" alt="Dice" style="width: 500px; height: 465px" width="500" height="465"><h2 id="understanding-zero-and-one-in-probability"><strong>Understanding Zero and One in Probability</strong></h2><ul><li><strong>Probability of 0:</strong> Consider an event like drawing a red card from a standard deck of 52 blue cards. Since there are no red cards, the probability is 0, making it an impossible event.</li></ul><p></p><ul><li><strong>Probability of 1: </strong>If you were to draw a card from a deck containing all red cards, the probability of drawing a red card would be 1, indicating certainty.</li></ul><h3><strong>Example 1: Drawing from a Deck of Blue Cards</strong></h3><ul><li><strong>Event</strong>: Drawing a red card from blue cards.</li><li><strong>Probability</strong>: 0 (Impossible).</li></ul><img src="https://tutorchase-production.s3.eu-west-2.amazonaws.com/8e4e37bc-f603-4d1a-804f-d58bb66e4e04-file.png" alt="Blue Deck of Cards" style="width: 500px; height: 285px" width="500" height="285"><h3><strong>Example 2: Drawing from a Deck of Red Cards</strong></h3><ul><li><strong>Event</strong>: Drawing a red card.</li><li><strong>Probability</strong>: 1 (Certain).</li></ul><img src="https://tutorchase-production.s3.eu-west-2.amazonaws.com/20fad189-43ea-4687-9b1b-01be637cc037-file.png" alt="Red Cards" style="width: 500px; height: 505px" width="500" height="505"><h2 id="visualising-probability"><strong>Visualising Probability</strong></h2><ul><li>Drawing an ace from a deck of cards.</li><li><strong>Total number of aces in a deck</strong>: 4.</li><li><strong>Total cards in a deck</strong>: 52.</li></ul><p></p>$P(\text{Ace}) = \frac{4}{52} = \frac{1}{13}$<p></p><p>Plotting this on the probability scale, we see that P(Ace) is closer to 0 than to 1, indicating a relatively low probability.</p><h2 id="real-world-applications"><strong>Real-World Applications</strong></h2><p>Probability scale is not just a theoretical concept; it has real-world applications in:</p><ul><li><strong>Decision making</strong>: Helps in assessing risks and making informed decisions.</li><li><strong>Predictive models</strong>: Used in weather forecasting, stock market analysis, and more.</li></ul><h2 id="practice-questions">Practice Questions</h2><h3><strong>Question 1</strong></h3><p>What is the probability of drawing a queen from a standard deck of cards?</p><p></p><h4><strong>Solution:</strong></h4><ul><li><strong>Total number of queens in a deck</strong>: 4.</li><li><strong>Total cards in a deck</strong>: 52.</li></ul><p></p>$P(\text{Queen}) = \frac{4}{52} = \frac{1}{13}$<p></p><h3><strong>Question 2</strong></h3><p>If a bag contains 3 red marbles and 7 blue marbles, what is the probability of drawing a red marble?</p><p></p><h4><strong>Solution:</strong></h4><p></p>$P(\text{Red Marble}) = \frac{3}{10}$