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Need help with probability

1. At a volleyball training at the gym, everybody gets three tries to hit a ball. The likelihood that Ludwig hits the ball at one strike is 0.4.
Let A denote the event that Ludwig meets at least once in three attempts.
a) Determine the complement event
b) Determine the probability of the complement event
c) Determine the likelihood that Ludwig hits the ball at least once in three tries.

My answers:
a) I answered that the complementary event was that Ludwig hit the ball only once, but the answer is tha he misses all the shots. How do you know that this is the complementary act? Is a little confusing.
b) As a result of the complement event, I calculated that the probability is (0.4) * (0.4) * (0.4). In fact, it is 0.22. How did they get that answer?
c) If I understand correctly, you write 1-P(complement event) .. but can anyone explain why you do this and how to get the answer 0.78?

2. Else exercises high jump and usually manage to jump 1.65m at 80% of all her attempts. Let A denote the event that Elsa can handle the height once at most, in two tries.

a) Determine the complement event A
b) Determine probability for complement event
c) Determine P (A), ie. the probability that Elsa can handle the height once at most in two tries. How do I come to that answer?

My answers:
a) Is the complementary event that Elsa can handle the height all the time (two)?
b) 4/5 * 4/5 = 16/25 Percentually, it should be 64%
c) What is the answer here? How do I know?

3. The probability that it rains seven days in a row in a town is 0.5%. What is the probability of having at least one rainy day in a week?
The event A is that he hits the ball at least once, so the complement is that he doesn't hit the ball at least once - i.e. he misses every time.

The probability of something happening at least once is (1 - the probability of it happening zero times). The chance of him missing the ball each time is 0.6 * 0.6 * 0.6 = 0.216 so the chance of at least one hit is 1 - 0.216 = 0.784
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manny103
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Original post by SerBronn
The event A is that he hits the ball at least once, so the complement is that he doesn't hit the ball at least once - i.e. he misses every time.

How do you know what the complement is, I read that the complement is something opposite to an event. So when they ask what the probability is for him to hit the ball once in three times, I think the opposite is that he only hit once or lower, because they wrote that he hits the ball atleast once in three attemps. So the complement to ''atleast'' is maximum or equal, no?

Original post by SerBronn
The chance of him missing the ball each time is 0.6 * 0.6 * 0.6 = 0.216 so the chance of at least one hit is 1 - 0.216 = 0.784


How did you get that ''1-0.216=probability of hitting once''? What's the rule we are following here? I'm not that well informed
You need to use mathematical definitions not vague textual analysis. The complement of a set of outcomes A is not(A). That's the set of all possible outcomes minus the ones in A.

What are all the possibilities of hitting and missing in three strikes? Well, they are HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM. Event A is that he hits it at least once. How many of those outcomes have at least one hit? Well all of them, except MMM.

The probability of all possible outcomes must sum to one so P[Event happening zero times] + P[Event happening one or more times] = 1

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