Independent Events Probability Calculations For F, G, And H
In the realm of probability theory, understanding the concept of independent events is crucial. Events are considered independent if the occurrence of one does not affect the probability of the other. This article delves into a specific scenario involving three independent events: F, G, and H. We are given the probabilities of events F and G, as well as the probability of the intersection of F and H. Our goal is to leverage this information to calculate various probabilities, including the intersections of these events.
We are given:
- P(F) = 2/3
- P(G) = 3/5
- P(F and H) = 14/45
Using this foundation, we will systematically determine the probabilities of:
- P(F and G)
- P(H)
- P(G and H)
- P(F and G and H)
This exploration will not only provide numerical answers but also illuminate the principles of independence and how they influence probability calculations. Let's dive into the calculations.
a. Determining P(F and G)
The cornerstone of calculating the probability of the intersection of independent events lies in the fundamental principle that the probability of two independent events occurring together is the product of their individual probabilities. In mathematical terms, if events F and G are independent, then:
P(F and G) = P(F) * P(G)
This formula is the bedrock of our calculation. We are explicitly told that events F, G, and H are independent. This independence is the key that unlocks the solution to this problem. It allows us to treat the occurrence of one event as completely unrelated to the occurrence of the others. This is a crucial simplification, as it bypasses the need to consider conditional probabilities or dependencies between the events.
Now, let's apply this principle to our specific scenario. We are given P(F) = 2/3 and P(G) = 3/5. Substituting these values into our formula, we get:
P(F and G) = (2/3) * (3/5)
To calculate this product, we multiply the numerators together and the denominators together:
P(F and G) = (2 * 3) / (3 * 5) P(F and G) = 6 / 15
However, it is essential to express probabilities in their lowest terms. Both the numerator and the denominator of 6/15 are divisible by 3. Dividing both by 3, we obtain:
P(F and G) = (6 ÷ 3) / (15 ÷ 3) P(F and G) = 2/5
Therefore, the probability of both events F and G occurring is 2/5. This result is a direct consequence of the independence of F and G and the multiplicative property that governs independent events.
In summary, by leveraging the independence of events F and G and applying the fundamental formula for the probability of independent events occurring together, we have successfully calculated P(F and G) to be 2/5. This exemplifies how understanding the core principles of probability allows us to solve complex problems in a straightforward manner.
b. Calculating P(H)
To determine the probability of event H, we once again leverage the principle of independence. We are given that events F and H are independent, and we also know P(F and H) = 14/45 and P(F) = 2/3. The key here is to utilize the formula for the probability of the intersection of independent events, which we introduced earlier:
P(F and H) = P(F) * P(H)
This formula is our primary tool for unlocking the value of P(H). It establishes a direct relationship between the probability of the joint occurrence of F and H and their individual probabilities. Since we know P(F and H) and P(F), we can rearrange this formula to solve for P(H).
Rearranging the formula, we divide both sides by P(F):
P(H) = P(F and H) / P(F)
Now, we substitute the given values:
P(H) = (14/45) / (2/3)
Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we rewrite the expression as:
P(H) = (14/45) * (3/2)
Next, we multiply the numerators and the denominators:
P(H) = (14 * 3) / (45 * 2) P(H) = 42 / 90
To express this fraction in its lowest terms, we need to find the greatest common divisor (GCD) of 42 and 90. Both numbers are divisible by 2, 3, and 6. The greatest of these is 6. Dividing both numerator and denominator by 6, we get:
P(H) = (42 ÷ 6) / (90 ÷ 6) P(H) = 7/15
Therefore, the probability of event H occurring is 7/15. This calculation highlights the power of leveraging the independence of events to isolate and determine individual probabilities when joint probabilities are known. The reciprocal relationship between division and multiplication with fractions is a crucial skill in this process, allowing us to manipulate equations and arrive at the solution.
In conclusion, by applying the fundamental principles of independent events and manipulating the relevant formula, we have successfully determined P(H) to be 7/15. This further reinforces the importance of understanding the multiplicative property of independent events in probability calculations.
c. Finding P(G and H)
Continuing our exploration of independent events, we now aim to calculate the probability of both events G and H occurring, denoted as P(G and H). As we've established, the key principle for independent events is that the probability of their joint occurrence is the product of their individual probabilities. Since events G and H are independent, we can apply the following formula:
P(G and H) = P(G) * P(H)
We already know P(G) = 3/5 from the problem statement. We also calculated P(H) = 7/15 in the previous section. Now, it's a matter of substituting these values into the formula:
P(G and H) = (3/5) * (7/15)
Multiplying the numerators and denominators, we get:
P(G and H) = (3 * 7) / (5 * 15) P(G and H) = 21 / 75
To express this fraction in its simplest form, we need to find the greatest common divisor (GCD) of 21 and 75. Both numbers are divisible by 3. Dividing both numerator and denominator by 3, we obtain:
P(G and H) = (21 ÷ 3) / (75 ÷ 3) P(G and H) = 7 / 25
Thus, the probability of both events G and H occurring is 7/25. This result underscores the direct and simple calculation that is possible when dealing with independent events. The multiplicative property allows us to move seamlessly from individual probabilities to joint probabilities.
In summary, by utilizing the independence of events G and H and applying the fundamental formula for the probability of the intersection of independent events, we have calculated P(G and H) to be 7/25. This continues to demonstrate how the core tenets of probability theory provide a clear framework for problem-solving.
d. Determining P(F and G and H)
Finally, we tackle the calculation of the probability of all three events F, G, and H occurring simultaneously, denoted as P(F and G and H). The fundamental principle of independence extends to multiple events. If events are mutually independent, the probability of their joint occurrence is the product of their individual probabilities. Therefore, for events F, G, and H, we have:
P(F and G and H) = P(F) * P(G) * P(H)
This formula is the cornerstone of our final calculation. It underscores the elegant simplicity that independence brings to probability calculations. Instead of grappling with complex conditional probabilities, we can directly multiply individual probabilities to find the probability of their joint occurrence.
We already have the individual probabilities:
- P(F) = 2/3
- P(G) = 3/5
- P(H) = 7/15
Substituting these values into the formula, we get:
P(F and G and H) = (2/3) * (3/5) * (7/15)
Now, we multiply the numerators together and the denominators together:
P(F and G and H) = (2 * 3 * 7) / (3 * 5 * 15) P(F and G and H) = 42 / 225
To express this fraction in its lowest terms, we need to find the greatest common divisor (GCD) of 42 and 225. The prime factorization of 42 is 2 * 3 * 7, and the prime factorization of 225 is 3 * 3 * 5 * 5. The only common factor is 3. Dividing both numerator and denominator by 3, we get:
P(F and G and H) = (42 ÷ 3) / (225 ÷ 3) P(F and G and H) = 14 / 75
Therefore, the probability of all three events F, G, and H occurring simultaneously is 14/75. This final calculation exemplifies the power and simplicity of the multiplicative property when dealing with independent events.
In conclusion, by applying the fundamental principles of independent events and extending the multiplicative property to three events, we have successfully calculated P(F and G and H) to be 14/75. This completes our analysis of the probabilities associated with these independent events.
Throughout this analysis, we've seen how the concept of independent events greatly simplifies probability calculations. The key takeaway is the multiplicative property: for independent events, the probability of their joint occurrence is the product of their individual probabilities. This principle allowed us to efficiently calculate P(F and G), P(H), P(G and H), and P(F and G and H). Understanding this concept is crucial for tackling more complex probability problems and is a foundational element in the broader field of statistics and data analysis.
Probability, Independent Events, Intersection, Probability Calculation, Mutually Independent, Joint Probability, Conditional Probabilities, Mathematics, Statistics
