Watch the video below then answer the questions that follow

Hi there. It's Adrian again and today we'll be learning about Probability. Imagine you are a contestant on a game show and you are given the choice of three doors: behind one door is a car, and behind the other doors, goats. Of course, you want to win the car. You pick one of the doors, say Door 3, and the host, who knows what's behind all of the doors, opens one of the other two doors, say Door 2, to reveal a goat. He then says to you, "Do you want to change your choice?" Is it to your advantage to switch to Door 1? The concepts of Probability can help you determine the answer. Why should we care about understanding probability? Probability forms the foundation for making inferences about a population. It helps us determine how likely it is to observe a random event. In statistics, we use probability to help us identify events that are surprising when they occur - these events have a low probability. Probability helps us answer the question "How often could something like this happen by chance?" Our answer to this question helps us connect our data with the real world. Let's compare the probability of being struck by lightning in any given year to the probability of winning Powerball's Jackpot. According to the National Weather Service, the probability of being struck by lightning in any given year is estimated to be 0.00000654. That's 6.54x10 to the -6. According to the Powerball rules in 2020, the probability of winning the Powerball is 0.00000000342. That's 3.42x10 to the -9. Both of these events are very unlikely to happen because their probabilities are close to zero. If we compare their relative likelihoods, it is 1,911 times more likely to be struck by lightning in a given year than it is for someone to win the Powerball. Now, before we can determine interesting probabilities like these, we need to understand five basic rules of probability. The first rule is that probabilities are always numbers between 0 and 1 (including 0 and 1). Probabilities can be expressed as fractions, decimals, or percents. If an event has a high likelihood of happening, it has a high probability, close to 1. If an event has a small likelihood of happening, it has a low probability, close to 0. When we use probability in a statement, we are using a number between 0 and 1 to indicate the likelihood of an event. Let's say you and your friends are discussing two basketball greats: Kobe Bryant and Michael Jordan. Kobe Bryant's career free throw percentage was 83.7%, or 0.837, and Michael Jordan's was 83.5%, or 0.835. If these two went head to head on free throw shots, who would have been more likely to make their first shot? Each athlete's probability is between 0 and 1, and both of their numbers are close to 1. The likelihood of either of them making their first free throw shot is quite high, but not certain. It's also important to know that the probability that something in the sample space will occur is 1. Another way to say this is the sum of the probabilities for all possible outcomes equals 1. That is, if S represents all events that are possible, the probability of S = 1; this is our second probability rule. Over the course of a player's NBA career, the number of times they attempt a free throw shot is recorded along with the number of times they are successful and the number of times they are not successful. The percentage of successful free throw attempts and the percentage of unsuccessful free throw attempts adds up to 100% or 1 and make what we call the sample space. The sample space is comprised of all simple events that could occur in a random experiment. The probability that a basketball player either makes or does not make a free throw attempt is 100% or 1. This leads us into the third rule about probability. If we know the probability that an event could happen, we can find the probability that the event could not happen by using what we know about events in the sample space. The probability that an event will not occur is 1 minus the probability that the event will occur. This third rule is known as the complement rule. For instance, if you have noticed that about 1 in 5 people around campus are wearing a fitness tracking device, what is the probability that someone on campus is not wearing a fitness tracking device? Well, the sample space is made up of people who either wear a fitness tracker or do not wear one. Since we have observed that about 1 in 5 people on campus wear a fitness tracker, we can say the probability of someone wearing a fitness tracker is about 20%. We can find the probability that someone is not wearing a fitness tracker by subtracting this probability from 1. So, there is an 80% chance that someone on campus is not wearing a fitness tracker. The fourth rule applies to situations when we want to find the probability that one event or another event could occur. In situations such as these, we need to first check if the two events are mutually exclusive. When two events share no outcomes in common, they are mutually exclusive. When two events share at least one outcome, they are not mutually exclusive. If two events A and B are mutually exclusive, then P(A or B) = P(A) +P(B). If two events, A and B, are not mutually exclusive, then P(A or B) = P(A) + P(B) - P(A and B). When two events A and B do share at least one outcome, the probability of A or B happening is found by adding the probability that A could occur and the probability that B could occur and then we must subtract the probability that they could both occur. If we do not subtract out the probability that both events could occur, then we will have a probability value that is too large because the shared outcomes were included twice. Note that if there were no shared outcomes between these two events, then there would be no need to subtract anything because P(both events) is equal to zero. Let's look at an example where two events share at least one outcome. I recently took a trip to the aquarium and studied the penguin exhibit for 20 minutes. During this time, I observed 27 penguins going about their various activities. I saw 21 enjoying sunbathing, 9 enjoying swimming, but 5 of the penguins enjoyed both sunbathing and swimming. Suppose the animal trainer selected one penguin at random from this group. What is the probability that it enjoyed sunbathing or swimming. Well, since some penguins enjoyed both sunbathing and swimming during my observation, these two events are not mutually exclusive. So, we will use this formula. The probability that a penguin enjoyed sunbathing is 21 out of 27. The probability that a penguin enjoyed swimming is 9 out of 27. The probability that a penguin enjoyed both sunbathing and swimming is 5 out of 27. Substitute these values into the formula and simplify. The probability the animal trainer randomly selects a penguin that enjoyed sunbathing or swimming is 25 out of 27 or 92.6%. The fifth rule applies to situations where we want to find the probability that one event and another event could occur. In situations such as these, we need to first check if the two events are independent. Two events are independent if knowing that one event occurred does not change the probability that the other event could occur. That means the probability A occurs given that B did occur is the same as the probability of A occurring by itself. When two events A and B are independent, the probability of A and B happening is found by multiplying the probability that A could occur with the probability that B could occur, which is represented as P(A and B) = P(A) times P(B). If two events, A and B, are associated with each other, then P(A and B) = P(A given B) times P(B). Let's come back to the discussion about Kobe Bryant and Michael Jordan. Imagine each of these basketball players attempted two free throws (we will assume that the free throws are independent). What is the probability that Michael Jordan made both of his free throws? Recall that Michael Jordan's career free throw percentage was 83.5%. Since the free throws are independent, we will multiply the probability that Michael's first free throw shot went in by the probability that Michael's second free throw shot went in. That is, 0.835 squared, which equals 69.7%. Since this probability is near to 1, it would not be surprising at all if Michael made both of his free throw shots. What is the probability that Kobe Bryant missed both of his free throw shots? Recall that Kobe's free throw percentage was 83.7%. Well, we need to find the probability that Kobe missed at least 1 free throw, and the complement rule can help us. We can subtract the probability that Kobe made a free throw from 1 to get the probability that Kobe missed a free throw. There is a 0.163 or 16.3% chance that Kobe missed one free throw shot and there is a 2.7% chance that he missed two free throw shots. Because this probability is close to zero, it would be very unusual if Kobe missed both free throw shots. When two events A and B are not independent, finding the probability that they both occur involves conditional probability because knowing that one event occurred changes the probability that the other event could occur. Remember the game show you were in earlier? And you had selected one of the three doors, Door 3, hoping to win a car? Then the host showed you that behind Door 2 was a goat. Well, now you need to decide if you will update your choice based on this new information and switch your selection to Door 1 or stay with your initial selection. Is it to your benefit to switch? In other words, given that we know the car is not behind Door 2 is the probability that Door 3 conceals the car less than the probability that Door 1 conceals the car? Let's apply what we know about probability to figure this one out. When you made your choice of Door 3, there was a 1 in 3 chance that the car was behind any one of the three doors. By randomly choosing Door 3, you left two doors unselected. Together, these two doors had a 2/3 chance of concealing the car and a 1/3 chance that they did not. Then the host opened Door 2 to reveal a goat. By revealing what was behind Door 2, the host provided you updated information to help you make your decision in a nonrandom way. Having more information about the location of the car, particularly that it is not behind Door 2, did three things for you: First, the reveal gave you additional nonrandom information about the location of the car; Second, the initial probability that the car was behind Door 1 or Door 2 was combined; and third, this combined probability was assigned to Door 1. So, there is still a 1 in 3 chance the car is behind Door 3 but now there is a 2 in 3 chance the car is behind Door 1. So, when the game show host asks you if you want to switch your choice from Door 3 to Door 1, you should make the switch - you will increase your probability of winning that car. We have discussed five basic rules of probability that can help you understand how surprising it might be if you observe an event or if you want to determine the probability that something might happen. As you learn more about statistics and the process of making inferential conclusions, these five rules of probability will help you connect your data with the real world in meaningful ways.

1. event with a probability close to zero is an event that is unlikely to happen. True or False
2. Which of the following are valid probabilities? (Select all that apply.) 0.99, -0.81, 1.75%, 1.01, 0.0000000004, 1/4
3. If the weather forecasts predict an 88% chance of rain today, what is the probability it will not rain today? 0.12, 0.12%, 0.88, 12