Questions: Events A and B are mutually exclusive. Suppose event A occurs with a probability of 0.15 and event B occurs with a probability of 0.4. Compute the following. (a) Compute the probability that B occurs but A does not occur. (b) Compute the probability that either A occurs without B occurring or B occurs without A occurring.

Events A and B are mutually exclusive. Suppose event A occurs with a probability of 0.15 and event B occurs with a probability of 0.4. Compute the following. (a) Compute the probability that B occurs but A does not occur. (b) Compute the probability that either A occurs without B occurring or B occurs without A occurring.
Transcript text: Events $A$ and $B$ are mutually exclusive. Suppose event $A$ occurs with probability 0.15 and event $B$ occurs with probability 0.4 . Compute the following. (a) Compute the probability that $B$ occurs but $A$ does not occur. (b) Compute the probability that either $A$ occurs without $B$ occurring or $B$ occurs without $A$ occurring.
failed

Solution

failed
failed

Solution Steps

To solve the given problems, we need to use the properties of mutually exclusive events and basic probability rules.

(a) Since events \(A\) and \(B\) are mutually exclusive, the probability that \(B\) occurs but \(A\) does not occur is simply the probability of \(B\) occurring, because \(A\) and \(B\) cannot occur together.

(b) The probability that either \(A\) occurs without \(B\) occurring or \(B\) occurs without \(A\) occurring is the sum of the probabilities of \(A\) and \(B\) occurring, since they are mutually exclusive.

Step 1: Calculate the Probability that \(B\) Occurs but \(A\) Does Not Occur

Since events \(A\) and \(B\) are mutually exclusive, the probability that \(B\) occurs but \(A\) does not occur is given by: \[ P(B \cap \neg A) = P(B) = 0.4 \]

Step 2: Calculate the Probability that Either \(A\) Occurs Without \(B\) Occurring or \(B\) Occurs Without \(A\) Occurring

The probability that either \(A\) occurs without \(B\) occurring or \(B\) occurs without \(A\) occurring is the sum of the probabilities of \(A\) and \(B\): \[ P(A \cap \neg B) + P(B \cap \neg A) = P(A) + P(B) = 0.15 + 0.4 = 0.55 \]

Final Answer

The answers to the sub-questions are:

  • (a) \(P(B \cap \neg A) = 0.4\)
  • (b) \(P(A \cap \neg B) + P(B \cap \neg A) = 0.55\)

Thus, the final answers are: \[ \boxed{P(B \cap \neg A) = 0.4} \] \[ \boxed{P(A \cap \neg B) + P(B \cap \neg A) = 0.55} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
For the normal distribution, the mean plus and minus two standard deviations will include about what percent of the observations?
failedAnswered
An artery has a diameter of 1.2 cm and blood flows through it with a velocity of 3.2 cm / s. If at some point there is plaque buildup, such that the artery has its diameter narrowed to 30% of its initial diameter. What velocity is the blood in the region with plaque buildup?
failedAnswered
During an instruction session on breast self-examination, the nurse would instruct a client to perform the exam at which time? On the 14th day of the menstrual period Immediately before the menstrual period begins Three days after the menstrual period begins Right after the menstrual period ends
failedAnswered
EACH of these statements is TRUE of quickly inserting columns and rows in Excel EXCEPT: Click the number to the left of the data to select the row; right click and select Insert to place a row directly BELOW To insert a column before another column, highlight the column, right click, and select Insert Insert multiple rows by clicking and dragging the number of number of rows you want to enter Click the number to the left of the data to select the row; right click and select Insert to place a row directly ABOVE
failedAnswered
Why was the market no longer in long-run equilibrium? When the market price rose, this means that existing firms were able to make positive economic profits. Ah, I see. A higher market price means that the firms who were making hand sanitizer before the increase in demand were able to earn positive economic profits. But, initially, how did the market reach the new equilibrium at Q1*? There was an increase in supply as new firms entered the market in search of positive economic profits. There was an increase in the quantity supplied as existing firms increased the amount of hand sanitizer they were producing.