Answer :
All right, let's solve the problem step by step.
We are given:
- \( P(C) = 0.5 \)
- \( P(D) = 0.3 \)
- Events \( C \) and \( D \) are independent.
### Step 1: Probability \( P(D) \)
The probability \( P(D) \) is simply given as 0.3.
### Step 2: Probability \( P(C \text{ and } D) \)
Since events \( C \) and \( D \) are independent, the probability of both events occurring simultaneously (i.e., \( P(C \text{ and } D) \)) is the product of their individual probabilities:
[tex]\[ P(C \text{ and } D) = P(C) \times P(D) = 0.5 \times 0.3 = 0.15 \][/tex]
### Step 3: Probability \( P(C \text{ or } D) \)
The probability of either event \( C \) or event \( D \) occurring (i.e., \( P(C \text{ or } D) \)) can be found using the formula:
[tex]\[ P(C \text{ or } D) = P(C) + P(D) - P(C \text{ and } D) \][/tex]
Substituting the given values:
[tex]\[ P(C \text{ or } D) = 0.5 + 0.3 - 0.15 = 0.65 \][/tex]
### Step 4: Conditional Probability \( P(C \mid D) \)
The conditional probability \( P(C \mid D) \) is the probability of \( C \) occurring given that \( D \) has occurred. This is calculated using the formula:
[tex]\[ P(C \mid D) = \frac{P(C \text{ and } D)}{P(D)} \][/tex]
Substituting the known values:
[tex]\[ P(C \mid D) = \frac{0.15}{0.3} = 0.5 \][/tex]
### Step 5: Probability \( P(\text{neither } C \text{ nor } D) \)
The probability that neither \( C \) nor \( D \) occurs is the complement of the probability that either \( C \) or \( D \) occurs. Hence:
[tex]\[ P(\text{neither } C \text{ nor } D) = 1 - P(C \text{ or } D) \][/tex]
Using the value from Step 3:
[tex]\[ P(\text{neither } C \text{ nor } D) = 1 - 0.65 = 0.35 \][/tex]
To summarize:
- \( P(D) \) is \( 0.3 \)
- \( P(C \text{ and } D) \) is \( 0.15 \)
- \( P(C \text{ or } D) \) is \( 0.65 \)
- \( P(C \mid D) \) is \( 0.5 \)
- [tex]\( P(\text{neither } C \text{ nor } D) \)[/tex] is [tex]\( 0.35 \)[/tex]
We are given:
- \( P(C) = 0.5 \)
- \( P(D) = 0.3 \)
- Events \( C \) and \( D \) are independent.
### Step 1: Probability \( P(D) \)
The probability \( P(D) \) is simply given as 0.3.
### Step 2: Probability \( P(C \text{ and } D) \)
Since events \( C \) and \( D \) are independent, the probability of both events occurring simultaneously (i.e., \( P(C \text{ and } D) \)) is the product of their individual probabilities:
[tex]\[ P(C \text{ and } D) = P(C) \times P(D) = 0.5 \times 0.3 = 0.15 \][/tex]
### Step 3: Probability \( P(C \text{ or } D) \)
The probability of either event \( C \) or event \( D \) occurring (i.e., \( P(C \text{ or } D) \)) can be found using the formula:
[tex]\[ P(C \text{ or } D) = P(C) + P(D) - P(C \text{ and } D) \][/tex]
Substituting the given values:
[tex]\[ P(C \text{ or } D) = 0.5 + 0.3 - 0.15 = 0.65 \][/tex]
### Step 4: Conditional Probability \( P(C \mid D) \)
The conditional probability \( P(C \mid D) \) is the probability of \( C \) occurring given that \( D \) has occurred. This is calculated using the formula:
[tex]\[ P(C \mid D) = \frac{P(C \text{ and } D)}{P(D)} \][/tex]
Substituting the known values:
[tex]\[ P(C \mid D) = \frac{0.15}{0.3} = 0.5 \][/tex]
### Step 5: Probability \( P(\text{neither } C \text{ nor } D) \)
The probability that neither \( C \) nor \( D \) occurs is the complement of the probability that either \( C \) or \( D \) occurs. Hence:
[tex]\[ P(\text{neither } C \text{ nor } D) = 1 - P(C \text{ or } D) \][/tex]
Using the value from Step 3:
[tex]\[ P(\text{neither } C \text{ nor } D) = 1 - 0.65 = 0.35 \][/tex]
To summarize:
- \( P(D) \) is \( 0.3 \)
- \( P(C \text{ and } D) \) is \( 0.15 \)
- \( P(C \text{ or } D) \) is \( 0.65 \)
- \( P(C \mid D) \) is \( 0.5 \)
- [tex]\( P(\text{neither } C \text{ nor } D) \)[/tex] is [tex]\( 0.35 \)[/tex]
