When working with probability, it is important to understand the relationship between the joint probability of two events and the individual probabilities of those events. One important rule to keep in mind is that the joint probability of two events occurring together is always greater than or equal to the sum of their individual probabilities minus 1.
For example, consider the case where P(E) = 0.9 and P(F) = 0.8. Using this rule, we can show that P(EF) >= 0.7. This can be represented mathematically as:
P(EF) >= P(E) + P(F) − 1
Where P(EF) is the joint probability of events E and F occurring together, P(E) is the probability of event E occurring, and P(F) is the probability of event F occurring.
This rule is based on the fact that joint probability represents the probability of two events occurring together and it is always greater than or equal to the probability of either of the events happening individually.
In general, this rule can be applied to any two events and can help to understand the relationship between joint and individual probabilities. It's important to note that this relationship holds true for any two events regardless of their independence or dependence.
P(EF) represents the probability of both events E and F occurring. P(E) represents the probability of event E occurring, and P(F) represents the probability of event F occurring.
Using the general rule that P(EF) >= P(E) + P(F) − 1, we can substitute the given values to show that:
P(EF) >= 0.9 + 0.8 - 1 = 0.7
This is because the probability of two events occurring together (joint probability) is always greater than or equal to the sum of their individual probabilities minus 1.
It is also possible to show this using the formula P(EF) = P(E) * P(F|E) , where P(F|E) is the probability of event F given that event E has occurred. As P(F|E) >=0 , the product P(E) * P(F|E) is greater or equal to P(E) * 0 which is P(E) and P(EF) >= P(E)
Therefore, P(EF) >= 0.7 when P(E) = 0.9 and P(F) = 0.8.
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