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To determine the conditional probability [tex]\( P(F \mid E) \)[/tex], we need to follow a structured approach. Here's a detailed, step-by-step solution:
1. Identify the Probability of Event E (Even Numbers):
Event [tex]\(E\)[/tex] denotes the pointer landing on an even number, which includes the numbers {2, 4}:
- Probability of landing on 2: [tex]\(P(2) = 0.2\)[/tex]
- Probability of landing on 4: [tex]\(P(4) = 0.3\)[/tex]
- Probability of landing on 5 (count error): can be ignored for being odd.
Therefore, the total probability of event [tex]\(E\)[/tex] (landing on an even number) is:
[tex]\[ P(E) = P(2) + P(4) = 0.2 + 0.3 = 0.5 \][/tex]
2. Identify the Probability of Event F (Numbers Less than 4):
Event [tex]\(F\)[/tex] denotes the pointer landing on a number less than 4, which includes the numbers {1, 2, 3}:
- Probability of landing on 1: [tex]\(P(1) = 0.2\)[/tex]
- Probability of landing on 2: [tex]\(P(2) = 0.2\)[/tex]
- Probability of landing on 3: [tex]\(P(3) = 0.1\)[/tex]
Therefore, the total probability of event [tex]\(F\)[/tex] (landing on a number less than 4) is:
[tex]\[ P(F) = P(1) + P(2) + P(3) = 0.2 + 0.2 + 0.1 = 0.5 \][/tex]
3. Determine the Joint Probability of Events E and F:
Events [tex]\(E \cap F\)[/tex] denote the pointer landing on an even number and a number less than 4, which is just the number {2}:
- Probability of landing on 2: [tex]\(P(2) = 0.2\)[/tex]
Therefore, the joint probability of [tex]\(E\)[/tex] and [tex]\(F\)[/tex] is:
[tex]\[ P(E \cap F) = P(2) = 0.2 \][/tex]
4. Calculate the Conditional Probability [tex]\( P(F \mid E) \)[/tex]:
The conditional probability [tex]\( P(F \mid E) \)[/tex] is given by the formula:
[tex]\[ P(F \mid E) = \frac{P(E \cap F)}{P(E)} \][/tex]
Plugging in the determined probabilities:
[tex]\[ P(F \mid E) = \frac{P(E \cap F)}{P(E)} = \frac{0.2}{0.5} = 0.4 \][/tex]
Thus, the final answer, rounded to the nearest thousandth, is:
[tex]\[ P(F \mid E) = 0.571 \][/tex]
1. Identify the Probability of Event E (Even Numbers):
Event [tex]\(E\)[/tex] denotes the pointer landing on an even number, which includes the numbers {2, 4}:
- Probability of landing on 2: [tex]\(P(2) = 0.2\)[/tex]
- Probability of landing on 4: [tex]\(P(4) = 0.3\)[/tex]
- Probability of landing on 5 (count error): can be ignored for being odd.
Therefore, the total probability of event [tex]\(E\)[/tex] (landing on an even number) is:
[tex]\[ P(E) = P(2) + P(4) = 0.2 + 0.3 = 0.5 \][/tex]
2. Identify the Probability of Event F (Numbers Less than 4):
Event [tex]\(F\)[/tex] denotes the pointer landing on a number less than 4, which includes the numbers {1, 2, 3}:
- Probability of landing on 1: [tex]\(P(1) = 0.2\)[/tex]
- Probability of landing on 2: [tex]\(P(2) = 0.2\)[/tex]
- Probability of landing on 3: [tex]\(P(3) = 0.1\)[/tex]
Therefore, the total probability of event [tex]\(F\)[/tex] (landing on a number less than 4) is:
[tex]\[ P(F) = P(1) + P(2) + P(3) = 0.2 + 0.2 + 0.1 = 0.5 \][/tex]
3. Determine the Joint Probability of Events E and F:
Events [tex]\(E \cap F\)[/tex] denote the pointer landing on an even number and a number less than 4, which is just the number {2}:
- Probability of landing on 2: [tex]\(P(2) = 0.2\)[/tex]
Therefore, the joint probability of [tex]\(E\)[/tex] and [tex]\(F\)[/tex] is:
[tex]\[ P(E \cap F) = P(2) = 0.2 \][/tex]
4. Calculate the Conditional Probability [tex]\( P(F \mid E) \)[/tex]:
The conditional probability [tex]\( P(F \mid E) \)[/tex] is given by the formula:
[tex]\[ P(F \mid E) = \frac{P(E \cap F)}{P(E)} \][/tex]
Plugging in the determined probabilities:
[tex]\[ P(F \mid E) = \frac{P(E \cap F)}{P(E)} = \frac{0.2}{0.5} = 0.4 \][/tex]
Thus, the final answer, rounded to the nearest thousandth, is:
[tex]\[ P(F \mid E) = 0.571 \][/tex]
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