At a fabric store, the probability that a customer buys thread is 0.15. The probability that a customer buys a needle given that the customer buys thread is 0.25. Which statement is true? A. Buying a needle and buying a thread are independent events. B. Every customer who buys a thread also buys a needle. C. Buying a needle and buying a thread are dependent events. D. The probability that a customer buys a needle and thread is a 0.10.
The Correct Answer and Explanation is:
Correct Answer: C. Buying a needle and buying a thread are dependent events.
Explanation:
In probability, two events are said to be independent if the occurrence of one event does not affect the probability of the other. In this case, we are given:
- The probability that a customer buys thread is P(Thread) = 0.15
- The probability that a customer buys a needle given that they bought thread is P(Needle | Thread) = 0.25
To check whether the events are independent, compare the conditional probability P(Needle | Thread) with the unconditional probability P(Needle). However, the value for P(Needle) alone is not given. Still, we can determine dependency by using the definition of dependent events.
Two events A and B are dependent if P(B | A) ≠ P(B). In this scenario, P(Needle | Thread) = 0.25, and we cannot say this is equal to P(Needle) unless it is given or calculated.
However, we can deduce something further. The fact that P(Needle | Thread) = 0.25 is a conditional probability implies that the likelihood of buying a needle changes depending on whether the customer bought thread. That is the essence of dependence. If the events were independent, P(Needle | Thread) would equal P(Needle).
So the given information clearly shows that the probability of buying a needle is influenced by whether or not the customer buys thread. This is a classic example of dependent events.
Let us also rule out the other options:
- A is incorrect because the events are not independent.
- B is incorrect because only 25 percent of thread buyers also buy needles, not all.
- D is incorrect because P(Needle and Thread) = P(Thread) × P(Needle | Thread) = 0.15 × 0.25 = 0.0375, not 0.10.
Thus, the correct choice is C — the events are dependent.
