Question

chris is taking two classes this semester, english and american history. the probability he passes english is 0.50 and the probability he passes american history is 0.40. the probability he passes either english or american history is 0.60. what is the probability that he passes both classes? 0.4 + 0.6 - 0.5 0.5 + 0.4 - 0.6 0.5 × 0.4 0.5 + 0.4 - (0.5 × 0.4)

Explanation:

Step1: Recall Probability Formula

The formula for the probability of the intersection (both events) when we know the probability of the union (either event) is \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \). Wait, no, actually, the formula for the union is \( P(A \cup B)=P(A)+P(B)-P(A \cap B) \), so we can solve for \( P(A \cap B) \) as \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \). Here, let \( A \) be passing English (\( P(A) = 0.5 \)), \( B \) be passing American History (\( P(B)=0.4 \)), and \( P(A \cup B)=0.6 \). Wait, no, the question is asking for the probability he passes both classes, which is \( P(A \cap B) \). Using the formula \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \). Wait, but let's check the options. Wait, maybe I misread. Wait, the options: let's list the options:

1. \( 0.4 + 0.6 - 0.5 \) – no, that's not the formula.
2. \( 0.5 + 0.4 - 0.6 \) – let's see: \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \). If \( P(A)=0.5 \), \( P(B)=0.4 \), \( P(A \cup B)=0.6 \), then \( P(A \cap B)=0.5 + 0.4 - 0.6 \)? Wait, no, wait the problem says: "The probability he passes either English or American History is 0.60". So \( P(A \cup B)=0.6 \), \( P(A)=0.5 \), \( P(B)=0.4 \). Then \( P(A \cap B)=P(A)+P(B)-P(A \cup B)=0.5 + 0.4 - 0.6 \)? Wait, but let's check the options. Wait, the fourth option is \( 0.5 + 0.4 - (0.5 \times 0.4) \) – no, that's if they are independent, but we know the union is 0.6. Wait, no, the formula for independent events is \( P(A \cap B)=P(A)P(B) \), but here we have the union given. Wait, maybe I made a mistake. Wait, the question is: "What is the probability that he passes both classes?" So we need \( P(A \cap B) \). The formula is \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \). Given \( P(A)=0.5 \), \( P(B)=0.4 \), \( P(A \cup B)=0.6 \). So substituting, \( P(A \cap B)=0.5 + 0.4 - 0.6 \). Wait, but let's check the options. The third option is \( 0.5 + 0.4 - 0.6 \)? Wait, no, the third option (second from left? Wait the options are:

First option: \( 0.4 + 0.6 - 0.5 \)

Second option: \( 0.5 + 0.4 - 0.6 \)

Third option: \( 0.5 \times 0.4 \)

Fourth option: \( 0.5 + 0.4 - (0.5 \times 0.4) \)

Wait, no, the original image: let's parse the options again. The first option (leftmost) is \( 0.4 + 0.6 - 0.5 \)? No, wait the text:

First option: \( 0.4 + 0.6 - 0.5 \)

Second option: \( 0.5 + 0.4 - 0.6 \)

Third option: \( 0.5 \times 0.4 \)

Fourth option: \( 0.5 + 0.4 - (0.5 \times 0.4) \)

Wait, the problem states: "The probability he passes either English or American History is 0.60". So \( P(A \cup B)=0.6 \), \( P(A)=0.5 \) (pass English), \( P(B)=0.4 \) (pass American History). We need \( P(A \cap B) \). Using the formula \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \). So substituting the values: \( 0.5 + 0.4 - 0.6 \). Let's compute that: \( 0.5 + 0.4 = 0.9 \), \( 0.9 - 0.6 = 0.3 \). Alternatively, if we check the other options:

- \( 0.4 + 0.6 - 0.5 = 0.5 \) – not correct.
- \( 0.5 \times 0.4 = 0.2 \) – that's if independent, but we know the union is 0.6, so they are not independent.
- \( 0.5 + 0.4 - (0.5 \times 0.4)=0.5 + 0.4 - 0.2 = 0.7 \) – not correct.
- \( 0.5 + 0.4 - 0.6 = 0.3 \), which fits the formula.

Wait, but let's confirm the formula. The principle of inclusion - exclusion for two events: \( |A \cup B| = |A| + |B| - |A \cap B| \), so in probability terms, \( P(A \cup B)=P(A)+P(B)-P(A \cap B) \). Solving for \( P(A \cap B) \), we get \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \). So with \( P(A)=0.5 \), \( P(B)=0.4 \), \( P(A \cup B)=0.6 \), then \( P(A \cap B)=0.5 + 0.4 - 0.6 = 0.3 \). So the correct option is the one with \( 0.5 + 0.4 - 0.6 \), which is the second option (the third circle? Wait the options are:

First circle: \( 0.4 + 0.6 - 0.5 \)

Second circle: \( 0.5 + 0.4 - 0.6 \)

Third circle: \( 0.5 \times 0.4 \)

Fourth circle: \( 0.5 + 0.4 - (0.5 \times 0.4) \)

Wait, the second option (the second circle from the left? Wait the original image: the options are:

1. \( 0.4 + 0.6 - 0.5 \)

2. \( 0.5 + 0.4 - 0.6 \)

3. \( 0.5 \times 0.4 \)

4. \( 0.5 + 0.4 - (0.5 \times 0.4) \)

So the correct one is the second option: \( 0.5 + 0.4 - 0.6 \)

Step2: Verify with Formula

Using \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \), where \( P(A)=0.5 \), \( P(B)=0.4 \), \( P(A \cup B)=0.6 \). Plugging in, we get \( 0.5 + 0.4 - 0.6 \), which matches the second option.

Answer:

The second option (the option with \( 0.5 + 0.4 - 0.6 \))