Probability question regarding independent events, where am I wrong ?

In summary, the conversation discusses two independent events, E1 and E2, and the probabilities of correctly judging each event. The problem asks for the probability of correctly judging only one event. The first attempt at a solution incorrectly calculates this probability by adding the probabilities of correctly judging E1 and E2, without accounting for the possibility of correctly judging both events. The correct solution involves subtracting the probability of correctly judging both events, which is equal to the product of their individual probabilities, from the sum of their individual probabilities. Another way to approach the problem is to directly calculate the probability of correctly judging only one event by adding the probabilities of correctly judging E1 but not E2 and correctly judging E2 but not E1.
  • #1
nishantve1
76
1

Homework Statement



E1 and E2 are two independent events. The probabilities of an error in judgement by a person regarding E1 and E2 are 0.02 and 0.05 respectively. Find the probability that the person will take the correct decision regarding :
only one event

Homework Equations


Two events are independent if occurrence of one is not affected by the other and vice versa .
Let,
C1 be the probability the person judging first one right = (1-0.02) = 0.98
C2 be the probability the person judging the second one right = (1-0.05) = 0.95

The Attempt at a Solution



So if the person judges one event correctly this means the he either judges the first one correctly or the second one correctly
P(only one event) = P(C1 or C2) = P(C1 + C2) =P(C1) + P(C2) - P(C1).P(C2)
=0.999
Which is the wrong anwer . But what's wrong with the logic ?

The hint shows
P(only one event) =
P(E1 is judged correctly and E2 is not judged correctly or E2 is judged correctly and E1 is not judged correctly)

How are these two logics different I mean the first one also finds the probability of one event judgement to be true . Doesn't it ?

Correct answer : 0.068
 
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  • #2
P(C1 or C2) = P(C1) + P(C2) − P(C1)·P(C2)
This equation correctly computes the probability of C1 inclusive or C2, not the probability of C1 exclusive or C2, as the problem asks.

Think about it like this: C1 includes the "just C1" event as well as the "C1 and C2" event, and C2 includes the "just C2" event as well as the "C1 and C2" event. When you add their probabilities, you double-count the "C1 and C2" event. Because C1 and C2 are independent, the probability of "C1 and C2" is indeed given by P(C1)·P(C2), but subtracting it once from the sum means you're still single-counting the probability of "C1 and C2." You don't want to count it at all, so you need to subtract that term again. You could modify your formula to get the right answer:

P(C1 xor C2) = P(C1) + P(C2) − 2P(C1)·P(C2)
 
  • #3
As alternative solution, you can directly evaluate
P(E1 is judged correctly and E2 is not judged correctly or E2 is judged correctly and E1 is not judged correctly)
as
P(C1)(1-P(C2)) + (1-P(C1))P(C2)
As those options are mutually exclusive, you can directly add their probabilites.
 

Related to Probability question regarding independent events, where am I wrong ?

1. What is the difference between independent and dependent events in probability?

Independent events are events that do not affect each other's outcome. This means that the outcome of one event does not change the probability of the other event occurring. On the other hand, dependent events are events that do affect each other's outcome. The outcome of one event can change the probability of the other event occurring.

2. Can two independent events both have a probability of 0.5?

Yes, two independent events can both have a probability of 0.5. This means that the probability of one event occurring does not affect the probability of the other event occurring. For example, flipping a coin and rolling a die are two independent events that both have a probability of 0.5.

3. How do you calculate the probability of two independent events occurring together?

To calculate the probability of two independent events occurring together, you multiply the probabilities of each event. For example, if the probability of event A is 0.3 and the probability of event B is 0.5, the probability of both events occurring together is 0.3 x 0.5 = 0.15.

4. Can the probability of two independent events occurring together be greater than 1?

No, the probability of two independent events occurring together cannot be greater than 1. This is because the maximum probability of any event occurring is 1. If the probability of both events occurring together is greater than 1, it means that there is a greater than 100% chance of both events happening, which is not possible.

5. How do you determine if two events are independent or dependent?

You can determine if two events are independent or dependent by looking at the relationship between them. If the outcome of one event does not affect the outcome of the other event, they are independent. If the outcome of one event does affect the outcome of the other event, they are dependent. Additionally, you can also calculate the conditional probability of the second event given the first event. If the conditional probability is equal to the probability of the second event, then the events are independent. If the conditional probability is not equal to the probability of the second event, then the events are dependent.

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