8. (New Hare-Hunting Problem) In the woods, a hunter is shooting at a hare. The probability of success for his first shot is ½. If he misses the first shot, the probability of success for his second shot is 1/4. If he misses again, the hunter takes his last shot with probability of success 1/8. What is the probability that he will hit the hare?

Well, if the hunter misses the first shot, we can consider it as a "hare-raising" experience. Now, the probability of success for the second shot is 1/4. That's four times better than the first shot, which means the hunter is probably saying, "Hare we go again!"

But let's say the hunter misses again. Now, it's time for the last shot, and the probability of success is 1/8. At this point, the hunter must be feeling quite "hared" and really hoping for a miracle.

To find the overall probability that the hunter hits the hare, we need to multiply the individual probabilities together. So, it's 1/2 (for the first shot) multiplied by 1/4 (for the second shot) multiplied by 1/8 (for the third shot).

Using my expert mathematical skills, I can tell you that the probability of the hunter hitting the hare is (1/2) * (1/4) * (1/8) = 1/64.

So, it looks like the hunter better not quit his day job because the probability of him hitting that hare is as slim as the chances of finding a vegetarian lion. Good luck, hunter!

To find the probability that the hunter will hit the hare, we need to consider the probabilities of each possible sequence of shots.

Let's denote H as a successful shot and M as a missed shot. We have three possible sequences:

1. H: The hunter hits the hare on his first shot with a probability of 1/2.

2. M -> H: The hunter misses the first shot (with a probability of 1/2) and then hits the hare on his second shot (with a probability of 1/4). The probability of this sequence is (1/2) * (1/4) = 1/8.

3. M -> M -> H: The hunter misses the first shot (with a probability of 1/2), then misses the second shot (with a probability of 1/4), and finally hits the hare on his last shot (with a probability of 1/8). The probability of this sequence is (1/2) * (1/4) * (1/8) = 1/64.

To find the total probability, we sum up the probabilities of all three possible sequences: 1/2 + 1/8 + 1/64.

Simplifying, we get 32/64 + 8/64 + 1/64 = 41/64.

Therefore, the probability that the hunter will hit the hare is 41/64.

To find the probability that the hunter will hit the hare, we need to consider the different possibilities and calculate the probabilities for each scenario.

Let's break down the problem step by step:

1. The hunter has three shots in total: the first shot, the second shot (if the first shot is missed), and the third shot (if both the first and second shots are missed).

2. The probability of success for each shot is given: 1/2 for the first shot, 1/4 for the second shot, and 1/8 for the third shot.

To calculate the probability that the hunter will hit the hare, we need to consider all possible scenarios:

Scenario 1: The hunter hits the hare with the first shot.
In this scenario, the probability is simply 1/2 because the first shot is successful.

Scenario 2: The hunter misses the first shot but hits the hare with the second shot.
In this scenario, the probability is (1/2) * (1/4) = 1/8 because the hunter misses the first shot (probability of 1/2) and hits the hare with the second shot (probability of 1/4).

Scenario 3: The hunter misses both the first and second shots but hits the hare with the third shot.
In this scenario, the probability is (1/2) * (3/4) * (1/8) = 3/64 because the hunter misses the first shot (probability of 1/2), misses the second shot (probability of 3/4), and hits the hare with the third shot (probability of 1/8).

To calculate the overall probability that the hunter will hit the hare, we add up the probabilities for each scenario: 1/2 + 1/8 + 3/64 = 21/32.

Therefore, the probability that the hunter will hit the hare is 21/32.