Assume that Pr[E]=0.55,Pr[F]=0.55,Pr[G]=0.55,Pr[E∪F]=0.85,Pr[E∪G]=0.8, and Pr[F∪G]=0.75.
Find:
Pr[E' U F]; Pr[F' ∩ G]; and Pr[E ∩ G]
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To find the solutions to the given probabilities, we will use the following formulas:
1. Pr[A'] = 1 - Pr[A]
2. Pr[A ∪ B] = Pr[A] + Pr[B] - Pr[A ∩ B]
3. Pr[A ∩ B] = Pr[A] + Pr[B] - Pr[A ∪ B]
Let's solve each problem step-by-step:
1. Pr[E' U F]:
First, let's find Pr[E']. Using the formula (1), Pr[E'] = 1 - Pr[E] = 1 - 0.55 = 0.45.
Now, using the formula (2), Pr[E' U F] = Pr[E'] + Pr[F] - Pr[E' ∩ F].
Since E' and F are mutually exclusive (they cannot happen at the same time), Pr[E' ∩ F] = 0.
Therefore, Pr[E' U F] = Pr[E'] + Pr[F] = 0.45 + 0.55 = 1.
So, Pr[E' U F] = 1.
2. Pr[F' ∩ G]:
Similarly, let's find Pr[F']. Using formula (1), Pr[F'] = 1 - Pr[F] = 1 - 0.55 = 0.45.
Now, using formula (3), Pr[F' ∩ G] = Pr[F'] + Pr[G] - Pr[F' ∪ G].
Since F' and G are also mutually exclusive (they cannot happen at the same time), Pr[F' ∪ G] = 0.
Therefore, Pr[F' ∩ G] = Pr[F'] + Pr[G] = 0.45 + 0.55 = 1.
So, Pr[F' ∩ G] = 1.
3. Pr[E ∩ G]:
Using formula (3), Pr[E ∩ G] = Pr[E] + Pr[G] - Pr[E ∪ G].
Given that Pr[E] = 0.55, Pr[G] = 0.55, and Pr[E ∪ G] = 0.8, we can substitute these values:
Pr[E ∩ G] = 0.55 + 0.55 - 0.8 = 0.75.
So, Pr[E ∩ G] = 0.75.
Therefore, the solutions are:
Pr[E' U F] = 1
Pr[F' ∩ G] = 1
Pr[E ∩ G] = 0.75.
To find the values of Pr[E' U F], Pr[F' ∩ G], and Pr[E ∩ G], we need to understand basic probability concepts and apply appropriate set operations. Let me break down each calculation step-by-step:
1. Pr[E' U F]:
- Pr[E'] represents the probability of the complement of event E, i.e., the probability that event E does not occur. We can find this by subtracting Pr[E] from 1: Pr[E'] = 1 - 0.55 = 0.45.
- Pr[E' U F] denotes the probability that either event E does not occur or event F occurs. In set notation, it represents the union of the complement of E and event F.
- To calculate Pr[E' U F], we can add the probabilities of E' and F together and then subtract the probability of their intersection (Pr[E ∩ F]).
- Pr[E ∩ F] can be found using the formula: Pr[E' U F] = Pr[E'] + Pr[F] - Pr[E ∩ F].
- However, we do not have the value for Pr[E ∩ F] directly. So we need to find it using the inclusion-exclusion principle.
- The inclusion-exclusion principle states that for any two events A and B: Pr[A U B] = Pr[A] + Pr[B] - Pr[A ∩ B].
- In our case, we want to find Pr[E' U F]. We know Pr[E' U F] = Pr[E'] + Pr[F] - Pr[E ∩ F].
- Pr[E' U F] is given as 0.85, Pr[E'] is 0.45, and Pr[F] is 0.55. We can substitute these values in the equation:
0.85 = 0.45 + 0.55 - Pr[E ∩ F].
- Rearranging the equation, we find Pr[E ∩ F] = 0.15.
- Now, we can calculate Pr[E' U F] using the formula: Pr[E' U F] = Pr[E'] + Pr[F] - Pr[E ∩ F]:
Pr[E' U F] = 0.45 + 0.55 - 0.15 = 0.85.
2. Pr[F' ∩ G]:
- Pr[F'] represents the probability of the complement of event F, i.e., the probability that event F does not occur. We can find this by subtracting Pr[F] from 1: Pr[F'] = 1 - 0.55 = 0.45.
- Pr[F' ∩ G] denotes the probability that event F does not occur and event G occurs. In set notation, it represents the intersection of the complement of F and event G.
- To calculate Pr[F' ∩ G], we multiply the probabilities of F' and G together: Pr[F' ∩ G] = Pr[F'] * Pr[G] = 0.45 * 0.55 = 0.2475.
3. Pr[E ∩ G]:
- Pr[E ∩ G] denotes the probability that both events E and G occur. In set notation, it represents the intersection of events E and G.
- Unfortunately, we do not have direct information about Pr[E ∩ G]. But we can use the inclusion-exclusion principle to find it.
- The inclusion-exclusion principle states that: Pr[A U B] = Pr[A] + Pr[B] - Pr[A ∩ B].
- In our case, we know Pr[E ∪ G] = 0.8, Pr[E] = 0.55, and Pr[G] = 0.55. To find Pr[E ∩ G], we rearrange the equation:
Pr[E ∪ G] = Pr[E] + Pr[G] - Pr[E ∩ G].
- Substituting the given values, we get: 0.8 = 0.55 + 0.55 - Pr[E ∩ G].
- Solving for Pr[E ∩ G], we find: Pr[E ∩ G] = 0.3.
So, to summarize the results:
- Pr[E' U F] = 0.85
- Pr[F' ∩ G] = 0.2475
- Pr[E ∩ G] = 0.3