For any two events A and B, the probability that at least one of them occur is 0.6. If A and B occur simultaneously with a probability 0.3, then P(A') + P(B') is
1.1
Understanding the Probability Problem
The question asks us to find the sum of the probabilities of the complements of two events, denoted as P(A') + P(B'). We are given the probability that at least one of the events occurs, which is P(A ∪ B), and the probability that both events occur simultaneously, which is P(A ∩ B).
Given Information
- The probability that at least one event occurs is P(A ∪ B) = 0.6.
- The probability that both events occur simultaneously is P(A ∩ B) = 0.3.
Key Probability Formulas
We will use the following standard probability formulas:
- The formula for the union of two events: $$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$
- The formula for the complement of an event: $$ P(A') = 1 - P(A) $$ $$ P(B') = 1 - P(B) $$
- Therefore, the sum of complements is: $$ P(A') + P(B') = (1 - P(A)) + (1 - P(B)) = 2 - (P(A) + P(B)) $$
Step-by-Step Calculation
Step 1: Calculate the sum of individual probabilities, P(A) + P(B)
We start with the formula for the union of two events: $$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$ Substitute the given values: $$ 0.6 = P(A) + P(B) - 0.3 $$ Rearrange the equation to solve for P(A) + P(B): $$ P(A) + P(B) = 0.6 + 0.3 $$ $$ P(A) + P(B) = 0.9 $$
Step 2: Calculate the sum of the complements, P(A') + P(B')
Now, use the relationship derived earlier for the sum of complements: $$ P(A') + P(B') = 2 - (P(A) + P(B)) $$ Substitute the value of P(A) + P(B) calculated in Step 1: $$ P(A') + P(B') = 2 - 0.9 $$ $$ P(A') + P(B') = 1.1 $$
Conclusion
The probability that at least one of the events A or B occurs is 0.6, and the probability that both occur simultaneously is 0.3. Using these values, we calculated the sum of the probabilities of their complements, P(A') + P(B'), to be 1.1.
Important Questions from Multiplication Theorem of Events
The probability that A speaks truth is 4 / 5 while this probability for B is 3 / 4. The probability that they contradict each other when asked to speak on a fact is
A student appears for tests I, II and III. The student is considered successful if the passes in tests I, II or I, III or all the three. The probabilities of the student passing in test I, II and III are m, n and 1/2 respectively. If the probability of the student to be successful is 1/2, then which one of the following is correct?
If \(\rm P(A\cup B)=\dfrac{5}{6}, P(A\cap B)=\dfrac{1}{3}\:and\:P(\bar A)=\dfrac{1}{2}\) , then which of the following is/are correct?
1. A and B are independent events.
2. A and B are mutually exclusive events.
Select the correct answer using the code given below.
In a lottery of 10 tickets numbered 1 to 10, two tickets are drawn simultaneously. What is the probability that both the tickets drawn have prime numbers?
In a series of 3 one-day cricket matches between teams A and B of a college, the probability of team A winning or drawing are 1/3 and 1/6 respectively. If a win, loss or draw gives 2, 0 and 1 point respectively, then what is the probability that team A will score 5 points in the series?
