QUESTION 4 - 1 POINT
A marksman has a probability of 0.308 for hitting a certain long-range target. What is the probability that it takes 5 shots for the marksman to hit the long-range target? (Round your answer to three decimal places, if necessary.)
Provide your answer below:
Answer
Thus, the probability that it takes 5 shots for the marksman to hit the long-range target is approximately \(\boxed{0.071}\).
Step 1 ：This problem is about geometric distribution. The geometric distribution gives the probability that the first occurrence of success requires k independent trials, each with success probability p. If the probability of success on each trial is p, then the probability that the kth trial (out of k trials) is the first success is \((1-p)^{(k-1)}*p\).
Step 2 ：In this case, the probability of hitting the target is 0.308, and we want to find the probability that the first hit occurs on the 5th shot.
Step 3 ：We can plug these values into the formula and calculate the result. Let p = 0.308 and k = 5.
Step 4 ：The calculation gives us a probability of approximately 0.07062770499276798.
Step 5 ：Rounding to three decimal places, the final answer is 0.071.
Step 6 ：Thus, the probability that it takes 5 shots for the marksman to hit the long-range target is approximately \(\boxed{0.071}\).
