Probability of Complementary Events & Sample Space

Summary

This video explains basic probability concepts using a 6-sided die as an example. It covers the sample space, the probability of events A and B occurring, and the union and intersection of events. The video also introduces the complement of an event and demonstrates how to calculate probabilities for events and their complements. Additionally, it shows how to apply the formula for complementary probabilities in real-life scenarios, such as determining the probability of selecting a non-red marble from a bag. The video helps viewers grasp fundamental probability principles through clear examples and calculations.

Takeaways

• 😀 The sample space of a 6-sided die includes the outcomes {1, 2, 3, 4, 5, 6}.
• 😀 Event A includes outcomes less than or equal to 4, i.e., {1, 2, 3, 4}.
• 😀 Event B includes outcomes {3, 4, 5}.
• 😀 The probability of both Event A and Event B occurring is the intersection of A and B, which is {3, 4}, resulting in a probability of 1/3 or approximately 33%.
• 😀 The probability of either Event A or Event B occurring is the union of A and B, which includes outcomes {1, 2, 3, 4, 5}, giving a probability of 5/6 or approximately 83.33%.
• 😀 The complement of Event A includes outcomes {5, 6}, and the probability of the complement is 1/3.
• 😀 The probability of Event A occurring is 2/3 or approximately 66.67%.
• 😀 The complement rule states that the probability of an event’s complement is 1 minus the probability of the event itself (P(A^c) = 1 - P(A)).
• 😀 In the example with marbles, if the probability of selecting a red marble is 35%, the probability of selecting a marble that is not red is 65%.
• 😀 The complement rule can be applied in real-world scenarios, such as calculating the chance of drawing a non-red marble after knowing the chance of drawing a red one.

Q & A

What is the sample space for rolling a 6-sided die?

The sample space for rolling a 6-sided die consists of the six possible outcomes: {1, 2, 3, 4, 5, 6}.

What outcomes does event A include?

Event A includes all natural numbers less than or equal to 4, so the outcomes are {1, 2, 3, 4}.

What outcomes does event B include?

Event B includes the outcomes {3, 4, 5}.

What is the probability of both event A and event B occurring?

The probability of both event A and event B occurring is the intersection of A and B, which are the outcomes {3, 4}. There are 2 favorable outcomes out of 6 possible outcomes, so the probability is 2/6, which simplifies to 1/3, or approximately 33%.

How do you calculate the probability of event A or event B occurring?

The probability of event A or event B occurring is found by taking the union of A and B, which includes the outcomes {1, 2, 3, 4, 5}. There are 5 favorable outcomes out of 6 possible outcomes, so the probability is 5/6, which is approximately 83.33%.

What is the complement of event A, and how is its probability calculated?

The complement of event A includes all outcomes that are not part of A. Since A includes {1, 2, 3, 4}, the complement consists of {5, 6}. The probability of the complement of event A is 2 favorable outcomes out of 6, so the probability is 2/6, or 1/3.

What is the probability of event A occurring?

The probability of event A occurring is the number of favorable outcomes (4 outcomes: {1, 2, 3, 4}) divided by the total number of possible outcomes (6). So, the probability is 4/6, which simplifies to 2/3.

What relationship exists between the probability of event A and its complement?

The probability of event A plus the probability of its complement always equals 1. For event A, the probability is 2/3, and for its complement, the probability is 1/3. Therefore, 2/3 + 1/3 = 1.

If the probability of selecting a red marble is 35%, what is the probability of selecting a marble that is not red?

The probability of selecting a marble that is not red is the complement of the probability of selecting a red marble. So, it is 1 - 0.35 = 0.65, or 65%.

Why is the formula P(A complement) = 1 - P(A) important?

The formula P(A complement) = 1 - P(A) is useful for calculating the probability of the complement of an event. It ensures that the total probability of an event and its complement adds up to 1, providing a quick way to determine the likelihood of the complement when the event's probability is known.