CHAPTER 3. COUNTING AND PROBABILITY Example 3.2.7. Joe is about to take a 10 question multiple-choice quiz. He has not studied for the quiz, so he will randomly select one of the 5 choices in each question. What we plan to observe is not the sequence of answers, but rather the sequence of successes and failures in his guessing effort. 99 a) How many different outcomes are there in the sample space8 b) What is the probability of each outcome? e) What is the probability of getting 1 correct and 9 wrong answers? Tf we write each outcome as a sequence of S's and F's, for success and failure, respectively, each outcome will have exactly 10 letters and in each position there are two choices. For example, one outcome may be written as SSSSFFSSFS, whereas another one may be written as FSSSFSSFSF. If we break down the task of selecting an outcome into 10 steps, namely the 10 of the selections of S or F, the product principle tells us that there are 2 x ...x 2 = 210 = 1,024 outcomes in the sample space. 10 times At this point we may be tempted to say that the probability of each outcome is but this is wrong because the outcomes are not equally 1,024 likely. In fact, in each step the probability of S is smaller than the probability of F. For each question there are 5 choices, only one of which is correct. So, when Joe guesses randomly, the probability of success is 0.2 and the probability of failure is 0.8, in each question. We need something like the product principle for probabilities in order to deal with this. And in fact, there is such a principle: Product Principle for Probabilities Suppose a random experiment can be broken down into a se- quence of steps, each one being a random experiment on its own. Assume moreover that the probability assignments of the differ- ent steps are independent of each other. Then the probability of an outcome in the whole random experiment is the product of the probabilities of the individual steps. 24. Refer to Example 3.2.7 on page 99, and find the probabilities of the following events: a) Getting at most 6 correct answers. b) Getting at least 7 correct answers.
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.
Question 24
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
