Understanding and Solving Probabilities

Probability is the measure of the likelihood that an event will occur. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. The formula for probability is:

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

Worked Example

Problem: What is the probability of rolling a sum of 7 with two six-sided dice?

1. Identify favorable outcomes: The pairs that give a sum of 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 favorable outcomes.
2. Determine total outcomes: Each die has 6 faces, so when rolling two dice, the total number of outcomes is:

$$6 \times 6 = 36$$

3. Calculate probability: Using the probability formula:

$$P(\text{sum of 7}) = \frac{6}{36} = \frac{1}{6}$$

Key Questions

• What is the probability of drawing an ace from a standard deck of cards?
• If you flip a coin three times, what is the probability of getting at least one head?
• How do you calculate the probability of independent events occurring?
• What are complementary events in probability?

Keywords

• Probability
• Favorable outcomes
• Total outcomes
• Independent events