Probability is a fundamental concept in mathematics, quantifying how likely it is that an event will occur. This section introduces the standard notation used to express probabilities, helping students understand how to calculate and interpret these values in various contexts.

What is Probability Notation?

Probability notation is a symbolic way to represent the likelihood of events. It's essential for communicating mathematical ideas clearly and concisely.

• $P(A)$: Represents the probability of event A happening.
• $P(A')$: Denotes the probability of event A not happening, also known as the complement of A.

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Interpreting Probability Values

The probability of any event ranges from 0 to 1, where:

• 0 indicates an impossible event.
• 1 signifies a certain event.
• Values between 0 and 1 represent the likelihood of an event occurring, with numbers closer to 1 indicating a higher probability.

Example 1: Calculating Basic Probability

Consider a simple experiment where we flip a fair coin once.

• Event A: The coin lands on heads.
• To find $P(A)$, we identify the number of ways A can happen and divide by the total number of outcomes.

Total outcomes: 2 (heads or tails) Favourable outcomes for A: 1 (heads)

This means there is a 50% chance of the coin landing on heads.

The Complement Rule

The complement rule is a fundamental principle that helps us find the probability of an event not occurring.

• Formula: $P(A') = 1 - P(A)$
• This rule is particularly useful when it is easier to calculate the probability of the event not happening than the event itself.

Example 2: Using the Complement Rule

If the probability of raining tomorrow is 0.3, what is the probability it will not rain?

• Given: $P(\text{Rain}) = 0.3$
• To find: $P(\text{Rain}')$

Using the complement rule:

Thus, there is a 70% chance it will not rain tomorrow.

Applying Probability Notation in Different Contexts

Probability notation is versatile and can be applied to various scenarios, from simple experiments like coin flips to more complex situations involving multiple events.

Multiple Coin Flips

When flipping a coin twice, the probability notation helps us organise and calculate the likelihood of different outcomes.

• Event A: Getting two heads in a row.
• Total outcomes: 4 (HH, HT, TH, TT)
• Favourable outcomes for A: 1 (HH)

Drawing Cards from a Deck

In a standard deck of 52 cards, probability notation allows us to calculate the chances of drawing specific cards.

• Event B: Drawing an ace.
• Total outcomes: 52
• Favourable outcomes for B: 4 (one ace of each suit)

Practice Questions

To solidify your understanding of probability notation, try solving these problems using the concepts discussed above.

Question 1:

What is the probability of rolling a 5 on a fair six-sided die?

Solution:

With six possible outcomes and only one favourable outcome, $P(\text{rolling a 5}) = \dfrac{1}{6}$.

Question 2:

If the probability of an event B is 0.45, what is (P(B'))?

Solution:

Using the complement rule, $P(B') = 1 - P(B) = 1 - 0.45 = 0.55$.

Question 3:

In a bag of 30 marbles, 10 are red. What is the probability of randomly selecting a red marble?

Solution:

With 30 total marbles and 10 red marbles, $P(\text{red marble}) = \dfrac{10}{30} = \dfrac{1}{3}$.