This problem hopes to show you the difference between independent events, where you use P(A and B) = P(A)⋅⋅P(B) and dependent events where you use P(A and B) = P(A)⋅⋅P(B given A). Note: Type your arithmetic in the answer box to keep your answers exact. For example, if the answer is (200/1000)*(199/999) then type this into the answer box. (a) Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that the manufacture has made sure that 20 percent of the boxes produced have the secret decoder ring. The other 80 percent produced have a different gift inside. If you randomly select two boxes of cereal, what is the probability that BOTH of them have the secret decoder ring? Solution: On this problem, since this is out of all the boxes manufactured, we can assume that the probability of getting the decoder ring in the second box is the same as the probability of getting the decoder ring in the first box, because there are many many boxes manufactured. When 1 box is selected that has the decoder ring, it does not reduce the probability of getting a decoder ring the second time. The probability of getting a decoder ring the first time is 20 percent and the probability of getting a decoder ring the second time is 20 percent. So these two events are independent. So we just multiply the two percents together. (b) Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that 2 of the 10 boxes on the shelf have the secret decoder ring. The other 8 boxes on the shelf have a different gift inside. If you randomly select two boxes of cereal from the shelf to purchase, what is the probability that BOTH of them have the secret decoder ring? Solution: Notice that the percentage of boxes with the decoder ring on this problem is 20 percent, which is the same as part a. On this problem however, since this is out of a given number of boxes (10 boxes), the probability of getting the decoder ring in the second box is NOT the same as the probability of getting the decoder ring in the first box. So these two events are dependent and we will get a different answer. The probability of getting a decoder ring the first time is 210210 but the probability of getting a decoder ring the second time is 2−110−12-110-1 or 1919 . So the answer is 210⋅19210⋅19 Just type this into the blank. MOM will calculate it. Now lets see what happens when we increase the sample size. (c) Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that 20 of the 100 boxes on the shelf have the secret decoder ring. The other 80 boxes on the shelf have a different gift inside. If you randomly select two boxes of cereal from the shelf to purchase, what is the probability that BOTH of them have the secret decoder ring? Notice that the percentage of boxes with the decoder ring on this problem is still 20 percent, which is the same as part a and part b. On this problem though, since the sample size has increased, your answer will be closer to part a. As your sample size gets larger, your answer gets closer to the answer we got in part a, where we assumed the events were independent. (d) Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that 2000 of the 10000 boxes on the shelf have the secret decoder ring. The other 8000 boxes on the shelf have a different gift inside. If you randomly select two boxes of cereal from the shelf to purchase, what is the probability that BOTH of them have the secret decoder ring? Notice that the percentage of boxes with the decoder ring on this problem is still 20 percent, which is the same as parts a, b and c. On this problem though, since the sample size is very large, your answer will be very close to part a. As your sample size gets larger, your answer gets closer to the answer we got in part a, where we assumed the events were independent.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
This problem hopes to show you the difference between independent
Note: Type your arithmetic in the answer box to keep your answers exact. For example, if the answer is (200/1000)*(199/999) then type this into the answer box.
(a) Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that the manufacture has made sure that 20 percent of the boxes produced have the secret decoder ring. The other 80 percent produced have a different gift inside. If you randomly select two boxes of cereal, what is the probability that BOTH of them have the secret decoder ring?
Solution: On this problem, since this is out of all the boxes manufactured, we can assume that the probability of getting the decoder ring in the second box is the same as the probability of getting the decoder ring in the first box, because there are many many boxes manufactured. When 1 box is selected that has the decoder ring, it does not reduce the probability of getting a decoder ring the second time. The probability of getting a decoder ring the first time is 20 percent and the probability of getting a decoder ring the second time is 20 percent. So these two events are independent. So we just multiply the two percents together.
(b) Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that 2 of the 10 boxes on the shelf have the secret decoder ring. The other 8 boxes on the shelf have a different gift inside. If you randomly select two boxes of cereal from the shelf to purchase, what is the probability that BOTH of them have the secret decoder ring?
Solution: Notice that the percentage of boxes with the decoder ring on this problem is 20 percent, which is the same as part a. On this problem however, since this is out of a given number of boxes (10 boxes), the probability of getting the decoder ring in the second box is NOT the same as the probability of getting the decoder ring in the first box. So these two events are dependent and we will get a different answer. The probability of getting a decoder ring the first time is 210210 but the probability of getting a decoder ring the second time is 2−110−12-110-1 or 1919 . So the answer is 210⋅19210⋅19 Just type this into the blank. MOM will calculate it.
Now lets see what happens when we increase the sample size.
(c) Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that 20 of the 100 boxes on the shelf have the secret decoder ring. The other 80 boxes on the shelf have a different gift inside. If you randomly select two boxes of cereal from the shelf to purchase, what is the probability that BOTH of them have the secret decoder ring?
Notice that the percentage of boxes with the decoder ring on this problem is still 20 percent, which is the same as part a and part b. On this problem though, since the sample size has increased, your answer will be closer to part a. As your sample size gets larger, your answer gets closer to the answer we got in part a, where we assumed the events were independent.
(d) Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that 2000 of the 10000 boxes on the shelf have the secret decoder ring. The other 8000 boxes on the shelf have a different gift inside. If you randomly select two boxes of cereal from the shelf to purchase, what is the probability that BOTH of them have the secret decoder ring?
Notice that the percentage of boxes with the decoder ring on this problem is still 20 percent, which is the same as parts a, b and c. On this problem though, since the sample size is very large, your answer will be very close to part a. As your sample size gets larger, your answer gets closer to the answer we got in part a, where we assumed the events were independent.
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