Question

13. events a and b are independent . suppose event a occurs with probability 0.57 and event b occurs with probability 0.62.

1. if event a or event b occurs, what is the probability that a occurs?

2. if B does not occur , what is the probability that a does not occur?

round your answer to atleast two decimal places if needed .

Answer 1

Answer 2

Given:

• Events A and B are independent.

• P(A) = 0.57

• P(B) = 0.62

• 
  

1). If event A or event B occurs, what is the probability that A occurs?

We need to find the conditional probability P(A | A ∪ B).

The formula for conditional probability is:

$$P(A\mathrm{\mid }A\cup B)=\frac{P(A\cap (A\cup B))}{P(A\cup B)}$$

Since $A\cap (A\cup B)=A$, this simplifies to:

$$P(A\mathrm{\mid }A\cup B)=\frac{P(A)}{P(A\cup B)}$$

First, calculate $P(A\cup B)$:

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Because A and B are independent:

$$P(A\cap B)=P(A)\cdot P(B)=0.57\times 0.62=0.3534$$

Now, calculate $P(A\cup B)$:

$$P(A\cup B)=0.57+0.62-0.3534=0.8366$$

Now, compute the conditional probability:

$$P(A\mathrm{\mid }A\cup B)=\frac{0.57}{0.8366}\approx 0.6813$$

Rounded to two decimal places:

$$P(A\mathrm{\mid }A\cup B)\approx 0.68$$

2). If B does not occur, what is the probability that A does not occur?

We need to find $P(\text{not }A\mathrm{\mid }\text{not }B)$.

Since A and B are independent, the occurrence of B (or not B) does not affect A. Therefore:

$$P(\text{not }A\mathrm{\mid }\text{not }B)=P(\text{not }A)=1-P(A)=1-0.57=0.43$$

Rounded to two decimal places:

$$P(\text{not }A\mathrm{\mid }\text{not }B)=0.43$$

 Answers is :-

1. The probability that A occurs given that A or B occurs is 0.68.

2. The probability that A does not occur given that B does not occur is 0.43

Answer 3

Let's break down this probability problem step-by-step:

Given Information:

• Events A and B are independent.

• P(A) = 0.57 (Probability of event A occurring)

• P(B) = 0.62 (Probability of event B occurring)

1. Probability that A occurs if A or B occurs:

We want to find P(A | A ∪ B), which is the probability that A occurs given that A or B occurs.

Using the conditional probability formula:

P(A | A ∪ B) = P(A ∩ (A ∪ B)) / P(A ∪ B)

Since A ∩ (A ∪ B) is just A, we have:

P(A | A ∪ B) = P(A) / P(A ∪ B)

To find P(A ∪ B), we use the formula for the probability of the union of two events:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Since A and B are independent, P(A ∩ B) = P(A)¹ * P(B)

P(A ∩ B) = 0.57 * 0.62 = 0.3534

Now, calculate P(A ∪ B):

P(A ∪ B) = 0.57 + 0.62 - 0.3534 = 0.8366

Finally, calculate P(A | A ∪ B):

P(A | A ∪ B) = 0.57 / 0.8366 ≈ 0.6813

Rounding to two decimal places:

P(A | A ∪ B) ≈ 0.68

2. Probability that A does not occur if B does not occur:

We want to find P(A' | B'), where A' is the complement of A (A does not occur) and B' is the complement of B (B does not occur).

Since A and B are independent, A' and B' are also independent. Therefore:

P(A' | B') = P(A')

We know that P(A') = 1 - P(A)

P(A') = 1 - 0.57 = 0.43

So, P(A' | B') = 0.43

Answer:

1. The probability that A occurs if A or B occurs is approximately 0.68.

2. If B does not occur, the probability that A does not occur is 0.43.