QuestionDetermine if events A and B are independent given $P(-A)=0.60$ and $P(B)=0.30$ . Choose the correct option.

Studdy Solution

STEP 1

Assumptions1. The probability of event not A, denoted as $(-A)$ , is0.60. The probability of event B, denoted as $(B)$ , is0.303. Events A and B are independent if the probability of both events occurring is equal to the product of their individual probabilities

STEP 2

First, we need to find the probability of event A. Since we know that the probability of not A is0.60, we can use the fact that the sum of the probabilities of an event and its complement is1.
$(A) =1 -(-A)$

STEP 3

Now, plug in the given value for $(-A)$ to calculate $(A)$ .
$(A) =1 -0.60$

STEP 4

Calculate the probability of event A.
$(A) =1 -0.60 =0.40$

STEP 5

Now, we need to check each of the provided options to see which one makes A and B independent events. Recall that two events A and B are independent if and only if $(A \, and \, B) =(A)(B)$ .

STEP 6

Check option A. If $(A \, or \, B) =0.90$ , then $(A \, and \, B) =(A) +(B) -(A \, or \, B)$ .

STEP 7

Plug in the values for $(A)$ , $(B)$ , and $(A \, or \, B)$ to calculate $(A \, and \, B)$ .
$(A \, and \, B) =0.40 +0.30 -0.90$

STEP 8

Calculate $(A \, and \, B)$ .
$(A \, and \, B) =0.40 +0.30 -0.90 = -0.20$

STEP 9

Since probabilities cannot be negative, option A is not correct.

STEP 10

Check option B. If $(A \, and \, B) =0.18$ , then we need to check if this is equal to $(A)(B)$ .

STEP 11

Calculate $(A)(B)$ .
$(A)(B) =0.40 \times0.30$

STEP 12

Calculate the product of $(A)$ and $(B)$ .
$(A)(B) =0.40 \times0.30 =0.12$

STEP 13

Since $(A \, and \, B) \neq(A)(B)$ , option B is not correct.

STEP 14

Check option C. If $(A \, and \, B) =0$ , then we need to check if this is equal to $(A)(B)$ .

STEP 15

Since we already calculated $(A)(B) =0.12$ in step12, option C is not correct.

STEP 16

Check option D. If $(A \, or \, B) =0.18$ , then $(A \, and \, B) =(A) +(B) -(A \, or \, B)$ .

STEP 17

Plug in the values for $(A)$ , $(B)$ , and $(A \, or \, B)$ to calculate $(A \, and \, B)$ .
$(A \, and \, B) =0.40 +0.30 -0.$

STEP 18

Calculate $(A \, and \, B)$ .
$(A \, and \, B) =0.40 +0.30 -0.18 =0.52$

STEP 19

Since $(A \, and \, B) \neq(A)(B)$ , option D is not correct.
Therefore, none of the options make A and B independent events.

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