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Probability is a way to measure how likely an event is to happen. Imagine you toss a coin. You know it can land either on Heads or Tails. Probability helps us understand the chance of getting heads or tails when you toss the coin.
Before we dive deeper, let's learn some important words:
Probability tells us how likely an event is, using numbers between 0 and 1. If an event is sure to happen, its probability is 1. If it cannot happen at all, its probability is 0.
Let's understand events and outcomes with examples.
An outcome is a single possible result. For example, when rolling a six-sided die, the outcomes are the numbers 1, 2, 3, 4, 5, and 6.
An event can be one or more outcomes grouped together. For example, "rolling an even number" is an event that includes outcomes 2, 4, and 6.
Events can be:
Let's look at a diagram showing the sample space for rolling a die and an event of rolling an even number:
To find the probability of an event, we use this simple formula:
Here,
The probability value is always between 0 and 1.
Let's see how to use this formula with different events:
| Event | Favorable Outcomes (n(E)) | Total Outcomes (n(S)) | Probability \(P(E) = \frac{n(E)}{n(S)}\) |
|---|---|---|---|
| Tossing a Head (Coin) | 1 (Head) | 2 (Head, Tail) | \(\frac{1}{2}\) |
| Rolling an Even Number (Die) | 3 (2, 4, 6) | 6 (1 to 6) | \(\frac{3}{6} = \frac{1}{2}\) |
| Drawing a Red Ball (Bag with 5 Red, 3 Blue) | 5 | 8 (5 + 3) | \(\frac{5}{8}\) |
Step 1: Identify the sample space. For a coin toss, the outcomes are Head (H) and Tail (T), so \(n(S) = 2\).
Step 2: Identify favorable outcomes. Getting a head is one outcome, so \(n(E) = 1\).
Step 3: Use the formula:
\[ P(\text{Head}) = \frac{n(E)}{n(S)} = \frac{1}{2} \]
Answer: The probability of getting a head is \(\frac{1}{2}\).
Step 1: Find total balls: \(5 + 3 + 2 = 10\), so \(n(S) = 10\).
Step 2: Favorable outcomes are red balls: \(n(E) = 5\).
Step 3: Calculate probability:
\[ P(\text{Red ball}) = \frac{5}{10} = \frac{1}{2} \]
Answer: The probability of drawing a red ball is \(\frac{1}{2}\).
Step 1: Total outcomes on a die are 6 (numbers 1 to 6), so \(n(S) = 6\).
Step 2: Even numbers are 2, 4, and 6, so \(n(E) = 3\).
Step 3: Calculate probability:
\[ P(\text{Even number}) = \frac{3}{6} = \frac{1}{2} \]
Answer: The probability of rolling an even number is \(\frac{1}{2}\).
Step 1: Total balls: \(4 + 6 + 10 = 20\), so \(n(S) = 20\).
Step 2: Number of green balls (unfavorable outcomes) = 4.
Step 3: Favorable outcomes (not green) = total - green = \(20 - 4 = 16\).
Step 4: Calculate probability:
\[ P(\text{Not green}) = \frac{16}{20} = \frac{4}{5} \]
Answer: The probability of not drawing a green ball is \(\frac{4}{5}\).
Step 1: Total cards in deck: 52, so \(n(S) = 52\).
Step 2: Number of Kings in deck: 4 (King of Hearts, Diamonds, Clubs, Spades), so \(n(E) = 4\).
Step 3: Calculate probability:
\[ P(\text{King}) = \frac{4}{52} = \frac{1}{13} \]
Answer: The probability of drawing a King is \(\frac{1}{13}\).
When to use: At the start of every probability problem to avoid errors.
When to use: When calculating the probability of "not" events is easier than direct counting.
When to use: For compound events or multi-step problems.
When to use: During the last step of probability calculation.
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