Peluang • Part 15: Peluang Kejadian Majemuk - Peluang Kejadian Bersyarat

Summary

In this video, we delve into conditional probability and how it applies to dependent events, where one event influences the outcome of another. The formula for conditional probability, P(A|B) = P(A ∩ B) / P(B), is explained in detail. Using an example with two dice, the speaker demonstrates how to calculate conditional probabilities based on specific conditions. Viewers are guided step-by-step through the process, emphasizing the importance of identifying the right events and applying the formula correctly. The video concludes with a call to explore further content and engage with questions or feedback.

Takeaways

• 😀 Introduction to the topic of conditional probability in compound events.
• 😀 Conditional probability refers to the probability of event A occurring given that event B has already occurred.
• 😀 When two events A and B are not independent, they influence each other, meaning the occurrence of one affects the probability of the other.
• 😀 The formula for conditional probability is P(A | B) = P(A ∩ B) / P(B), where P(A ∩ B) is the probability of both events occurring, and P(B) is the probability of event B.
• 😀 It is crucial to remember that P(B) cannot be zero in conditional probability calculations, as division by zero is undefined.
• 😀 An example is provided with two dice being rolled together, where the sample space is formed by all possible outcomes from the dice rolls.
• 😀 The problem involves calculating the probability that the first die shows 1, given that the sum of the dice is less than 4.
• 😀 In solving this, the relevant outcomes are identified from the sample space that match the condition (sum of dice less than 4), and the calculation proceeds using the formula for conditional probability.
• 😀 Another example involves calculating the probability of the sum of the dice being greater than 9, given that the first die shows an odd number.
• 😀 The key takeaway is to correctly identify the events and apply the formula for conditional probability by isolating the necessary intersections and ensuring accurate counting of sample space outcomes.

Q & A

What is conditional probability?

Conditional probability refers to the probability of an event occurring given that another event has already occurred. It's calculated using the formula P(A|B) = P(A ∩ B) / P(B), where A and B are two events, and P(A|B) represents the probability of A happening given that B has happened.

What does the notation P(A|B) mean in conditional probability?

P(A|B) means the probability of event A occurring, given that event B has already occurred. It's read as 'the probability of A given B'.

What happens if events A and B are not independent?

If events A and B are not independent, it means that the occurrence of one event affects the probability of the other event. In this case, conditional probability applies, and the probability of A given B is calculated using the formula P(A|B) = P(A ∩ B) / P(B).

What does 'P(A ∩ B)' represent?

'P(A ∩ B)' represents the probability of both events A and B occurring simultaneously. It's also referred to as the intersection of events A and B.

How do you calculate conditional probability when event B occurs first?

When event B occurs first, conditional probability is calculated using the formula P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.

What is the key difference between P(A|B) and P(B|A)?

P(A|B) represents the probability of event A occurring given that event B has occurred, while P(B|A) represents the probability of event B occurring given that event A has occurred. They are calculated using different formulas, with the roles of A and B reversed.

In the dice problem, what was the sample space for throwing two dice?

The sample space for throwing two dice consists of all possible outcomes of the two dice rolls. Each die has 6 faces, so the total number of outcomes is 6 × 6 = 36. These outcomes are represented as pairs (x, y), where x and y are the numbers rolled on the first and second dice, respectively.

How do you calculate the conditional probability of the first die showing '1' given the sum is less than 4?

To calculate this, first identify the outcomes where the sum of the dice is less than 4. These outcomes are (1, 1), (1, 2), (2, 1), and (2, 2). Then, identify the outcomes where the first die shows '1' and the sum is less than 4, which are (1, 1) and (1, 2). The conditional probability is then P(A|B) = P(A ∩ B) / P(B), which equals 2/3.

In the second dice problem, what was the condition given for event B?

In the second dice problem, event B was defined as the first die showing an odd number. This means the possible outcomes for the first die were 1, 3, and 5, and the second die could show any number from 1 to 6.

What was the conditional probability of the sum of the dice being greater than 9, given the first die shows an odd number?

The conditional probability of the sum being greater than 9, given that the first die shows an odd number, was calculated by finding the outcomes where the first die is odd (1, 3, 5) and the sum is greater than 9. The relevant outcomes were (5, 5), (5, 6), (3, 6), (1, 10). The probability was found to be 1/9.