Question

josie chooses a shirt from her drawer at random. let m be the event that she chooses a shirt with stripes. let n be the event that she chooses a shirt with buttons. events m and n have the following probabilities.

• ( p(m) = 0.375 )
• ( p(n) = 0.25 )
• ( p(m \text{ and } n) = 0.125 )

complete the sentence.
events m and n must dropdown independent because options: ( 0.375 - 0.25 = 0.125 ), ( 0.375 cdot 0.25 = 0.125 ), ( 0.375 - 0.25
eq 0.125 ), ( 0.375 cdot 0.25
eq 0.125 )

Explanation:

Step1: Recall independence rule

For independent events, \( P(M \text{ and } N) = P(M) \times P(N) \).

Step2: Calculate \( P(M) \times P(N) \)

Given \( P(M) = 0.375 \), \( P(N) = 0.25 \).
Compute \( 0.375 \times 0.25 \):
\( 0.375 \times 0.25 = 0.09375 \)? Wait, no—wait, \( 0.375 = \frac{3}{8} \), \( 0.25 = \frac{1}{4} \), so \( \frac{3}{8} \times \frac{1}{4} = \frac{3}{32} = 0.09375 \)? Wait, no, the problem's \( P(M \text{ and } N) = 0.125 \). Wait, wait, let's recalculate: \( 0.375 \times 0.25 = 0.09375 \)? No, wait, \( 0.375 \times 0.25 \): \( 0.375 \times 0.25 = (3/8) \times (1/4) = 3/32 = 0.09375 \). But \( P(M \text{ and } N) = 0.125 \). Wait, no—wait, maybe I miscalculated. Wait, \( 0.375 \times 0.25 \): \( 0.375 \times 0.25 = 0.09375 \), but \( 0.375 \times 0.25 = 0.09375 \), and \( P(M \text{ and } N) = 0.125 \). Wait, no, the options: let's check the options. Wait, the options are:

1. \( 0.375 - 0.25 = 0.125 \) → \( 0.125 = 0.125 \), but subtraction isn't the rule for independence.
2. \( 0.375 \cdot 0.25 = 0.125 \) → Calculate \( 0.375 \times 0.25 \): \( 0.375 \times 0.25 = 0.09375 \)? Wait, no, \( 0.375 \times 0.25 = 0.09375 \), but \( 0.125 \) is \( 1/8 \). Wait, maybe a typo? Wait, \( 0.375 = 3/8 \), \( 0.25 = 1/4 \), \( 3/8 \times 1/4 = 3/32 = 0.09375 \). But \( P(M \text{ and } N) = 0.125 = 1/8 = 4/32 \). So \( 0.375 \times 0.25 = 0.09375 eq 0.125 \)? Wait, no—wait, the options: let's re-express. Wait, \( 0.375 \times 0.25 = 0.09375 \), but the option says \( 0.375 \cdot 0.25 = 0.125 \). Wait, maybe I made a mistake. Wait, \( 0.375 \times 0.25 \): \( 0.375 \times 0.25 = 0.09375 \), which is not 0.125. But the option says \( 0.375 \cdot 0.25 = 0.125 \). Wait, no—wait, maybe the numbers are different. Wait, the problem says \( P(M) = 0.375 \), \( P(N) = 0.25 \), \( P(M \text{ and } N) = 0.125 \). Let's check the independence rule: if independent, \( P(M \cap N) = P(M)P(N) \). So compute \( P(M)P(N) = 0.375 \times 0.25 = 0.09375 \). But \( P(M \cap N) = 0.125 \). Wait, that would mean they are not independent? But the options: wait, maybe I miscalculated \( 0.375 \times 0.25 \). Wait, \( 0.375 \times 0.25 \): \( 0.375 \times 0.25 = (3/8) \times (1/4) = 3/32 = 0.09375 \). But \( 0.125 = 1/8 = 4/32 \). So \( 0.375 \times 0.25 = 0.09375 eq 0.125 \). But the option says \( 0.375 \cdot 0.25 = 0.125 \). Wait, maybe the problem has a typo, or I'm missing something. Wait, no—wait, \( 0.375 \times 0.25 = 0.09375 \), but the option says \( 0.375 \cdot 0.25 = 0.125 \). Wait, maybe the numbers are \( P(M) = 0.5 \), \( P(N) = 0.25 \), then \( 0.5 \times 0.25 = 0.125 \). But the problem says \( P(M) = 0.375 \). Wait, maybe I made a mistake. Wait, let's check the options again. The first dropdown is "be" or "not be" independent. Let's recall: two events are independent if \( P(M \cap N) = P(M)P(N) \). So compute \( P(M)P(N) = 0.375 \times 0.25 = 0.09375 \). \( P(M \cap N) = 0.125 \). Since \( 0.09375 eq 0.125 \), they are not independent? But the options for the reason: wait, the options are:

- \( 0.375 - 0.25 = 0.125 \) → \( 0.125 = 0.125 \), but subtraction is irrelevant.
- \( 0.375 \cdot 0.25 = 0.125 \) → Let's compute \( 0.375 \times 0.25 \): \( 0.375 \times 0.25 = 0.09375 \), which is not 0.125. Wait, no—wait, \( 0.375 \times 0.25 = 0.09375 \), but \( 0.125 \) is \( 1/8 \). Wait, maybe the problem meant \( P(M) = 0.5 \)? No, the problem says 0.375. Wait, maybe I miscalculated \( 0.375 \times 0.25 \). Let's do decimal multiplication: 0.375 × 0.25. 375 × 25 = 9375, then divide by 1000 × 100 = 100000? Wait, no: 0.375 is 375/1000, 0.25 is 25/100. So (375/1000) × (25/100) = (375×25)/(1000×100) = 9375/100000 = 0.09375. Yes. So \( 0.375 \times 0.25 = 0.09375 eq 0.125 \). But the option says \( 0.375 \cdot 0.25 = 0.125 \). Wait, maybe the problem has a mistake, but let's check the options. Wait, the first part: "Events M and N must [be/not be] independent because [reason]". Let's re-express the independence rule: if \( P(M \cap N) = P(M)P(N) \), then independent. So compute \( P(M)P(N) = 0.375 \times 0.25 = 0.09375 \). \( P(M \cap N) = 0.125 \). Since \( 0.09375 eq 0.125 \), they are not independent? But the option for the reason: wait, the options are:

1. \( 0.375 - 0.25 = 0.125 \) → 0.125 = 0.125, but subtraction is not the rule.
2. \( 0.375 \cdot 0.25 = 0.125 \) → 0.09375 = 0.125? No.
3. \( 0.375 - 0.25 eq 0.125 \) → 0.125 ≠ 0.125? No, 0.375 - 0.25 = 0.125.
4. \( 0.375 \cdot 0.25 eq 0.125 \) → 0.09375 ≠ 0.125, which is true.

Wait, so the correct reason is \( 0.375 \cdot 0.25 eq 0.125 \), so events M and N are not independent because \( P(M) \times P(N) eq P(M \text{ and } N) \). Wait, but let's check again. Wait, the problem says "Events M and N must [be/not be] independent because [reason]". So the first dropdown: "not be", and the reason is \( 0.375 \cdot 0.25 eq 0.125 \). Wait, but let's confirm the independence rule. Two events are independent if the occurrence of one does not affect the probability of the other, which is mathematically \( P(M \cap N) = P(M)P(N) \). So if \( P(M)P(N) = P(M \cap N) \), they are independent; otherwise, not. Here, \( P(M)P(N) = 0.375 \times 0.25 = 0.09375 \), and \( P(M \cap N) = 0.125 \). Since \( 0.09375 eq 0.125 \), they are not independent, and the reason is \( 0.375 \cdot 0.25 eq 0.125 \).

Answer:

Events M and N must not be independent because \( 0.375 \cdot 0.25
eq 0.125 \).
(First dropdown: "not be"; second dropdown: \( 0.375 \cdot 0.25
eq 0.125 \))